Integral Calculus - Cheat Sheet
Your last-minute revision companion covering indefinite integrals, definite integrals, areas between curves, volumes of revolution, centroids, and moments. Quick formulas, key triggers, and exam-critical pitfalls — all on one sheet.
Indefinite Integration (Antiderivatives)
Formulas
∫ x^n dx = (x^(n+1))/(n+1) + C
Meaning
n = power; C = constant of integration (n ≠ −1)
Watch out
MUST add C for indefinite integrals; exponent rule: add 1 to power, divide by new power
When to use
Integrating any polynomial term with constant power
∫ (1/x) dx = ln|x| + C
Meaning
Natural logarithm of absolute value of x
Watch out
Must use absolute value |x|; if x < 0, ln gives domain error without absolute bars
When to use
When integrand is 1/x (e.g., ∫ dx/(5x) = (1/5)ln|x| + C)
∫ e^x dx = e^x + C
Meaning
e = Euler's number (~2.718)
Watch out
For e^(kx): ∫ e^(kx) dx = (1/k)e^(kx) + C; don't forget the 1/k factor
When to use
Exponential function with base e
∫ a^x dx = (a^x)/(ln a) + C
Meaning
a = constant base; a > 0, a ≠ 1
Watch out
Denominator is ln(a), NOT log(a); if a = e, formula reduces to e^x
When to use
Exponential with base other than e (e.g., 2^x, 10^x)
∫ sin x dx = −cos x + C
Meaning
Integral of sine is negative cosine
Watch out
NEGATIVE sign before cosine; ∫ sin(kx) dx = −(1/k)cos(kx) + C
When to use
Trigonometric sine function
∫ cos x dx = sin x + C
Meaning
Integral of cosine is sine (POSITIVE)
Watch out
POSITIVE sign; ∫ cos(kx) dx = (1/k)sin(kx) + C
When to use
Trigonometric cosine function
∫ tan x dx = −ln|cos x| + C = ln|sec x| + C
Meaning
Two equivalent forms; sec x = 1/cos x
Watch out
Either form acceptable; both equivalent via logarithm properties
When to use
Tangent function
∫ sec^2 x dx = tan x + C
Meaning
Secant squared integrates to tangent
Watch out
For ∫ sec^2(kx) dx = (1/k)tan(kx) + C
When to use
When you see sec²x or 1/cos²x
∫ csc^2 x dx = −cot x + C
Meaning
Cosecant squared integrates to NEGATIVE cotangent
Watch out
NEGATIVE sign; cot x = cos x / sin x
When to use
When you see csc²x or 1/sin²x
∫ sec x tan x dx = sec x + C
Meaning
Product of secant and tangent
Watch out
Watch for sec(kx)tan(kx): ∫ sec(kx)tan(kx) dx = (1/k)sec(kx) + C
When to use
When integrand is sec x tan x (recognize the pair)
∫ csc x cot x dx = −csc x + C
Meaning
Product of cosecant and cotangent
Watch out
NEGATIVE sign; ∫ csc(kx)cot(kx) dx = −(1/k)csc(kx) + C
When to use
When integrand is csc x cot x
∫ (1/√(1−x²)) dx = arcsin x + C
Meaning
Arcsine (inverse sine); domain: −1 < x < 1
Watch out
Domain: |x| < 1; for ∫ (1/√(a²−x²)) dx = arcsin(x/a) + C
When to use
When integrand has form 1/√(1−u²) (u = variable or function of x)
∫ (1/(1+x²)) dx = arctan x + C
Meaning
Arctangent (inverse tangent); domain: all real x
Watch out
For ∫ (1/(a²+x²)) dx = (1/a)arctan(x/a) + C; no domain restriction
When to use
When integrand is 1/(1+u²)
∫ (1/(x√(x²−1))) dx = arcsec|x| + C
Meaning
Arcsecant; domain: |x| > 1
Watch out
Absolute value critical; domain: |x| > 1
When to use
When integrand has form 1/(x√(x²−1))
Important facts
- Power rule: add 1 to exponent, divide by new exponent
- Always add +C for indefinite integrals (unless told otherwise)
- Linearity: ∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx
- Substitution (u-substitution) is the reverse of chain rule
- For 1/x, always use ln|x| (absolute value essential for negative x)
Key definitions
Antiderivative
Example
Antiderivative of f(x) = 2x is F(x) = x² + C
A function F(x) whose derivative is f(x); i.e., F'(x) = f(x). All antiderivatives differ by a constant.
Indefinite Integral
Example
∫ 3x² dx = x³ + C
The family of all antiderivatives of f(x), written ∫ f(x) dx = F(x) + C.
Constant of Integration (C)
Example
∫ cos x dx = sin x + C (C could be 0, 5, −3, etc.)
Arbitrary constant added to indefinite integrals to represent the family of antiderivatives.
Diagrams to know
- Graph of f(x) and its antiderivative F(x); slope = f(x)
- U-substitution: identify u, du, rewrite integrand, integrate, back-substitute
Integration Techniques
Formulas
u-substitution: ∫ f(g(x))·g'(x) dx = ∫ f(u) du (where u = g(x))
Meaning
u = inner function; du = u'(x) dx; substitute, integrate, replace u back
Watch out
MUST convert du from dx; don't forget back-substitution; check that du matches exactly
When to use
Chain rule in reverse; when integrand has composite function with its derivative present
Integration by Parts: ∫ u dv = uv − ∫ v du
Meaning
u, v are chosen parts; dv = integrand factor; v = ∫ dv; du = u'dx
Watch out
Choose u wisely; wrong choice leads to harder integral; may need parts twice; watch signs
When to use
Product of functions (e.g., xe^x, x sin x, x ln x); LIATE rule chooses u: Logarithm, Inverse trig, Algebra, Trig, Exponential (highest priority = u)
Partial Fractions: P(x)/Q(x) = A/(x−a) + B/(x−b) + ... (linear factors) or (Ax+B)/(x²+bx+c) (irreducible quadratic)
Meaning
Decompose rational function into sum of simpler fractions; A, B, ... constants found by matching coefficients or substitution
Watch out
If deg(P) ≥ deg(Q), do long division first; repeated factors: (x−a)² → A/(x−a) + B/(x−a)²; quadratic factors must be irreducible (discriminant < 0)
When to use
Integrating rational functions (polynomial/polynomial); degree of P < degree of Q
Important facts
- LIATE rule: choose u as Logarithm, Inverse trig, Algebra, Trig, Exponential (top priority)
- For products with trig/exponential, may need parts twice; watch for cycles
- Partial fractions: degree of numerator must be less than denominator; if not, use polynomial long division
- Check: ∫ u dv + ∫ v du should equal the original integrand (verification)
Key definitions
U-Substitution
Example
∫ (2x)(x²+1)³ dx: let u = x²+1, du = 2x dx → ∫ u³ du = u⁴/4 + C = (x²+1)⁴/4 + C
Technique replacing u = g(x) and du = g'(x) dx to simplify composite functions.
Integration by Parts
Example
∫ x sin x dx: u = x, dv = sin x dx → uv − ∫ v du = −x cos x + ∫ cos x dx = −x cos x + sin x + C
Method ∫ u dv = uv − ∫ v du for products; LIATE rule prioritizes choice of u.
Partial Fractions
Example
(5x+3)/[(x+1)(x−2)] = A/(x+1) + B/(x−2); solving: A = −8/3, B = 13/3
Decomposition of rational function into sum of simpler fractions with linear or irreducible quadratic denominators.
Diagrams to know
- LIATE priority ladder for u-substitution choice
- Flowchart: 'Can I factor denominator?' → partial fractions
Definite Integrals & Fundamental Theorem of Calculus
Formulas
∫_a^b f(x) dx = F(b) − F(a) (Fundamental Theorem of Calculus, Part 1)
Meaning
a, b = bounds; F(x) = antiderivative of f(x); evaluate F at upper bound minus lower bound
Watch out
Order matters: UPPER minus LOWER (not reversed); bounds are included as equality
When to use
Any definite integral; compute antiderivative, plug in bounds, subtract
∫_a^b f(x) dx = − ∫_b^a f(x) dx (Reversing Bounds)
Meaning
Swapping bounds reverses sign
Watch out
Negative sign appears when bounds reversed
When to use
Rearranging integral limits
∫_a^c f(x) dx = ∫_a^b f(x) dx + ∫_b^c f(x) dx (Additivity of Bounds)
Meaning
Split integral at intermediate point b (a < b < c)
Watch out
Order of bounds: must go left to right; b must lie strictly between a and c
When to use
When f(x) has discontinuity or change in formula within [a,c]
∫_a^b [f(x) ± g(x)] dx = ∫_a^b f(x) dx ± ∫_a^b g(x) dx (Linearity)
Meaning
Definite integral of sum = sum of definite integrals; same for scalar multiples
Watch out
Applies to + and − but not × or ÷
When to use
Any polynomial or sum of functions
∫_a^b k·f(x) dx = k ∫_a^b f(x) dx (Constant Multiple)
Meaning
Constant factor k pulls out of integral
Watch out
k is constant, not function of x
When to use
When integrand has constant multiplier
d/dx [∫_a^x f(t) dt] = f(x) (Fundamental Theorem of Calculus, Part 2)
Meaning
Derivative of integral (with variable upper limit) = integrand at that limit
Watch out
Must use chain rule if upper limit is g(x): d/dx [∫_a^(g(x)) f(t) dt] = f(g(x))·g'(x)
When to use
Differentiating integrals with variable bounds
Important facts
- Definite integral result is always a NUMBER, not a function
- ∫_a^a f(x) dx = 0 (same upper and lower bounds)
- If f(x) ≥ 0 on [a,b], then ∫_a^b f(x) dx ≥ 0
- If f(x) changes sign within interval, split at zeros to avoid cancellation of areas
Key definitions
Definite Integral
Example
∫_0^2 3x² dx = [x³]_0^2 = 8 − 0 = 8
∫_a^b f(x) dx represents the signed area under curve f(x) from x = a to x = b; value is a number (not a function).
Fundamental Theorem of Calculus
Example
Part 1: ∫_1^3 2x dx = [x²]_1^3 = 9 − 1 = 8. Part 2: d/dx ∫_0^x sin t dt = sin x
Part 1: ∫_a^b f(x) dx = F(b) − F(a) if F'(x) = f(x). Part 2: d/dx ∫_a^x f(t) dt = f(x).
Signed Area
Example
∫_0^π sin x dx = 2 (all above axis), but ∫_0^(2π) sin x dx = 0 (halves cancel)
Area above x-axis counts positive; area below x-axis counts negative.
Diagrams to know
- Graph: area under curve from a to b (shaded region); above x-axis = positive area
- Diagram showing signed area with parts above and below x-axis
Area Between Curves
Formulas
A = ∫_a^b [f(x) − g(x)] dx (Area between curves: x-integration, vertical strips)
Meaning
f(x) = upper curve, g(x) = lower curve; a, b = left, right bounds (x-coordinates of intersection)
Watch out
MUST determine which is upper/lower (test point or sketch); if roles reverse in interval, split integral; order: ALWAYS upper − lower (result ≥ 0)
When to use
When curves are functions of x; find intersection points (set f = g), integrate upper minus lower
A = ∫_c^d [x_right(y) − x_left(y)] dy (Area between curves: y-integration, horizontal strips)
Meaning
x_right(y) = rightmost curve, x_left(y) = leftmost curve; c, d = lower, upper y-bounds
Watch out
RIGHT minus LEFT (opposite of x-method); useful when dy integration simpler or vertical strips awkward
When to use
When easier to write x as function of y, or when curves are vertical/near-vertical; find intersection (y-values), integrate right minus left
Important facts
- Always sketch both curves to identify which is upper/lower
- Find ALL intersection points in the interval; if curves cross, split integral
- Area is ALWAYS non-negative: integrate |f−g| or split at crossings
- For horizontal strips (dy), curves must be invertible (solve x = f(y)); c ≤ y ≤ d is y-range of intersection
Key definitions
Area Between Curves
Example
Between y = x and y = x² from x = 0 to 1: A = ∫_0^1 (x − x²) dx = [x²/2 − x³/3]_0^1 = 1/2 − 1/3 = 1/6
Definite integral of the absolute difference between two functions over an interval; represents the total enclosed area.
Intersection Points
Example
y = x² and y = 2x intersect where x² = 2x → x(x−2) = 0 → x = 0 or x = 2
Points where two curves meet; found by solving f(x) = g(x) (or x_right = x_left for horizontal strips).
Diagrams to know
- Graph of two intersecting curves with shaded area between them; label upper/lower curves, intersection points, and bounds a, b
Volumes of Revolution
Formulas
V = π ∫_a^b [R(x)]² dx (Disk Method: revolve about x-axis)
Meaning
R(x) = radius (distance from x-axis to curve); a, b = x-bounds; disk has area πr² at each x
Watch out
R² not (∫R)²; radius = y-coordinate; if axis is NOT x-axis, R = distance to that axis (e.g., about y = c, use R = |f(x)−c|)
When to use
Revolving region bounded by y = f(x) ≥ 0, y = 0, x = a, x = b about the x-axis
V = π ∫_a^b [R_outer²(x) − R_inner²(x)] dx (Washer Method: revolve about x-axis with hole)
Meaning
R_outer = outer radius (top curve), R_inner = inner radius (bottom curve); both squared
Watch out
BOTH radii squared; π is common factor; R_inner² SUBTRACTED; check which curve is farther from axis
When to use
Revolving region between two curves (upper and lower) about the x-axis; upper curve must have larger radius
V = π ∫_c^d [R(y)]² dy (Disk Method: revolve about y-axis)
Meaning
R(y) = radius (distance from y-axis to curve x = f(y)); c, d = y-bounds
Watch out
Integrate with respect to y; radius = x-coordinate = f(y); washer version subtracts inner²
When to use
Revolving region bounded by x = f(y) ≥ 0, x = 0, y = c, y = d about the y-axis
V = 2π ∫_a^b x·f(x) dx (Shell Method: revolve about y-axis)
Meaning
x = radius of cylindrical shell; f(x) = height; shell volume = 2πrh; integrate along x
Watch out
Factor 2π is part of formula (not added separately); x·f(x) is the integrand; NOT [x·f(x)]²
When to use
Alternative to disk/washer for y-axis; especially useful when f(x) is hard to invert or bounds awkward for disk method
V = 2π ∫_c^d y·x(y) dy (Shell Method: revolve about x-axis)
Meaning
y = radius; x(y) = height; integrate with respect to y
Watch out
Radius = y (distance from x-axis); height = x(y); less common but valid alternative
When to use
Alternative for revolving about x-axis when washer method is harder
Important facts
- Always identify: axis of revolution, bounds, and which curve is which (outer/inner, upper/lower)
- Disk & washer: integrate along the axis of revolution (x for x-axis, y for y-axis)
- Shell method: always 2π∫rh; integrate perpendicular to axis (x for y-axis, y for x-axis)
- Volume is ALWAYS π × (something); check units: if in SI, volume in m³
- Washer: outer radius is MAX distance from axis; inner is MIN distance
Key definitions
Disk Method
Example
y = √x from 0 to 4 revolved about x-axis: V = π∫_0^4 (√x)² dx = π∫_0^4 x dx = π[x²/2]_0^4 = 8π
Volume by stacking circular disks perpendicular to axis of revolution; V = π∫R² da (where da is differential along axis).
Washer Method
Example
Between y = x² and y = 2x about x-axis (0 ≤ x ≤ 2): V = π∫_0^2 [(2x)² − (x²)²] dx
Volume by stacking washers (disks with holes) when region between two curves; V = π∫(R_outer² − R_inner²) da.
Shell Method
Example
y = x² from 0 to 2 revolved about y-axis: V = 2π∫_0^2 x·x² dx = 2π∫_0^2 x³ dx = 8π
Volume by summing cylindrical shells; V = 2π∫r·h da (r = radius, h = height, da = differential perpendicular to shell).
Diagrams to know
- Cross-section of solid at x (or y): show disk shape with R labeled
- Cylindrical shell 'unrolled': height h, circumference 2πr, thickness dr (or dy)
- 3D sketch: region in xy-plane and resulting solid of revolution
Centroids & Moments
Formulas
M_y = ∫ x·dA = ∫_a^b x·f(x) dx (First Moment about y-axis)
Meaning
M_y = weighted x-position; dA = infinitesimal area element (f(x) dx for vertical strip)
Watch out
Integrand is x (not x²); dA = f(x) dx for region under single curve; for washer, dA = [R_outer² − R_inner²] dx
When to use
Computing x-coordinate of centroid; represents 'rotational tendency' about y-axis
M_x = ∫ y·dA = ∫_a^b (1/2)[f(x)]² dx (First Moment about x-axis, vertical strip)
Meaning
M_x = weighted y-position; (1/2)f(x) is centroid of vertical strip (half height); multiply by height f(x) to get y-contribution
Watch out
Factor (1/2)f(x) is CRITICAL: centroid of strip is at y/2, so y_centroid_strip × area_strip = (f/2) × f dx = (1/2)f² dx; NOT just ∫ f(x) dx
When to use
Computing y-coordinate of centroid; vertical strip from y = 0 to y = f(x)
A = ∫_a^b f(x) dx (Area of region under curve)
Meaning
Total area from x = a to x = b under y = f(x)
Watch out
Simple definite integral; no x or y weighting
When to use
Denominator for centroid formulas; also for normalization
bar(x) = M_y / A = [∫_a^b x·f(x) dx] / [∫_a^b f(x) dx] (x-coordinate of centroid)
Meaning
x-bar = weighted average x-position over the region
Watch out
Numerator: x·f(x); denominator: f(x) only; don't confuse with moment formula
When to use
Finding horizontal position of geometric center
bar(y) = M_x / A = [∫_a^b (1/2)[f(x)]² dx] / [∫_a^b f(x) dx] (y-coordinate of centroid)
Meaning
y-bar = weighted average y-position; (1/2)f² in numerator accounts for strip's own centroid
Watch out
Numerator is (1/2)f² (NOT just f²); common mistake: using f(x) instead of (1/2)f²
When to use
Finding vertical position of geometric center
I_x = ∫ y² dA = ∫_a^b (1/3)[f(x)]³ dx (Second Moment / Moment of Inertia about x-axis, vertical strip)
Meaning
I_x = measure of resistance to rotation about x-axis; (1/3)f(x) is second moment per unit width for vertical strip
Watch out
Exponent 3 (not 2); (1/3)f³ comes from ∫_0^f y² dy = f³/3 for strip centroid; critical for engineering applications
When to use
Structural analysis (beams, bending); (1/3)f³ for vertical strip from y = 0 to y = f(x)
I_y = ∫ x² dA = ∫_a^b x²·f(x) dx (Second Moment / Moment of Inertia about y-axis)
Meaning
I_y = resistance to rotation about y-axis; integrate x²·dA
Watch out
Integrand is x²·f(x) (not x·f); different from M_y
When to use
Structural rotation about vertical axis
I_G = I_parallel − Ad² (Parallel Axis Theorem for composite shapes)
Meaning
I_G = centroidal moment of inertia; I_parallel = moment about any parallel axis; A = area; d = distance between axes
Watch out
Formula is I_parallel = I_G + Ad² (rearrange: I_G = I_parallel − Ad²); d is distance between axes; A is total area
When to use
Finding moment of inertia about centroid given I about another parallel axis (or vice versa)
Common values
Value
x̄ = b/3 from vertex (along median)
Symbol
Various; depends on orientation
Quantity
Centroid of triangle (base b, height h)
Value
ȳ = 4r/(3π) ≈ 0.424r from diameter
Symbol
Standard tabulation
Quantity
Centroid of semicircle (radius r, diameter on x-axis)
Value
I_base = (1/3)bh³
Symbol
I
Quantity
Second moment of rectangle (width b, height h) about base
Value
I_G = (1/12)bh³
Symbol
I_G
Quantity
Second moment of rectangle about centroid
Important facts
- Vertical strip dA = f(x) dx; centroid of strip is at x and y_strip = f(x)/2
- For M_x: integrate (1/2)f² (NOT just f); this is the most common mistake
- Centroid formula: x̄ and ȳ both involve division by total area A
- For composite shapes, split into simpler regions, find each centroid & I, then combine
- I_x units are [length]⁴ (e.g., m⁴ in SI); used extensively in structural design (beams, columns)
Key definitions
Centroid
Example
Centroid of rectangle from (0,0) to (4,2) is (2,1)
Geometric center of a region; point (x̄, ȳ) where the area is 'balanced'; x̄ = M_y/A, ȳ = M_x/A.
First Moment
Example
M_y = ∫_0^2 x·x² dx = ∫_0^2 x³ dx; centroid x-coordinate = M_y / A
M = ∫ (coordinate)·dA; M_x about x-axis, M_y about y-axis; first moment divided by area gives centroid coordinate.
Second Moment (Moment of Inertia)
Example
I_x = ∫_0^2 (1/3)(x²)³ dx for curve y = x² from 0 to 2
I = ∫ (coordinate)² dA; I_x about x-axis, I_y about y-axis; measures resistance to rotation.
Parallel Axis Theorem
Example
Rectangle 2×4: I_G about centroid = (1/12)×2×4³ = 10.67; about corner 2 units away: I = 10.67 + 8×4 = 42.67
Relates moment of inertia about centroid (I_G) to moment about any parallel axis: I_parallel = I_G + Ad².
Diagrams to know
- Vertical strip: show width dx at position x, height f(x); label centroid at (x, f(x)/2)
- Region divided into composite parts: each with centroid (x_i, y_i) and area A_i; overall centroid at weighted average
- Centroid and moment of inertia axes for standard shapes (rectangle, triangle, semicircle)
Arc Length
Formulas
L = ∫_a^b √[1 + (dy/dx)²] dx (Arc Length: curve y = f(x) from x = a to x = b)
Meaning
dy/dx = f'(x); integrand √[1 + (f')²] ds element; L = total distance along curve
Watch out
Integrand is √[1 + (dy/dx)²], NOT √[(dy/dx)²]; must sum along the curve (length is always positive)
When to use
Finding length of a curve given as y = f(x); differentiate f(x) to get dy/dx
L = ∫_c^d √[1 + (dx/dy)²] dy (Arc Length: curve x = g(y) from y = c to y = d)
Meaning
dx/dy = g'(y); integrate along y-axis
Watch out
√[1 + (dx/dy)²]; same structure, different variable
When to use
When curve is naturally expressed as x = g(y), or when dx/dy is simpler than dy/dx
L = ∫_a^b √[x'(t)² + y'(t)²] dt (Arc Length: parametric curve x = x(t), y = y(t), t ∈ [a,b])
Meaning
x'(t) = dx/dt; y'(t) = dy/dt; integrand is speed along curve
Watch out
Both x'² and y'² appear under square root (not subtracted); t bounds, NOT x bounds
When to use
Parametric curves (e.g., projectile motion, circle parametrization); avoids explicit y = f(x)
Important facts
- Arc length integrals are often difficult (no closed form); may require numerical methods or special techniques
- For straight line from (a,c) to (b,d): L = √[(b−a)² + (d−c)²] (Pythagorean distance)
- Arc length element: ds = √[1 + (dy/dx)²] dx = √[(dx)² + (dy)²]
- Parametric form useful for curves like circles, ellipses, cycloids
Key definitions
Arc Length
Example
Length of y = x^(3/2) from x = 0 to x = 1: L = ∫_0^1 √[1 + (3x/2)²] dx
Total distance measured along a curve between two points; always positive; computed by integrating the differential arc length ds.
Diagrams to know
- Curve y = f(x) with small arc element ds labeled; show right triangle with dx, dy, and hypotenuse ds
Must remember
- POWER RULE: ∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ −1); exponent increases by 1, divide by new exponent
- CONSTANT OF INTEGRATION: ALWAYS add +C to indefinite integrals (if omitted, exam deduction)
- DEFINITE INTEGRAL: ∫_a^b f(x) dx = F(b) − F(a) (UPPER value minus LOWER value, not reversed)
- AREA BETWEEN CURVES: A = ∫_a^b [f_upper − f_lower] dx (ALWAYS upper MINUS lower; bounds = x-values at intersections)
- CENTROID y-COORDINATE TRAP: ȳ = (1/2) ∫ [f(x)]² dx / A (numerator is (1/2)f², NOT just f²; this is the #1 mistake in centroid problems)
- DISK METHOD: V = π ∫ [R(x)]² dx; radius is squared, π factor included; if washer, subtract inner²: π ∫ (R_outer² − R_inner²) dx
- SHELL METHOD: V = 2π ∫ x·f(x) dx (about y-axis) or V = 2π ∫ y·x(y) dy (about x-axis); NOT [x·f]², just x·f
- INTEGRATION BY PARTS: ∫ u dv = uv − ∫ v du; LIATE rule chooses u (Logarithm > Inverse trig > Algebra > Trig > Exponential)
- MOMENT OF INERTIA: I_x = ∫ (1/3)[f(x)]³ dx (factor 1/3, exponent 3); commonly confused with M_x = ∫ (1/2)f² dx
- TRIGONOMETRIC INTEGRALS: ∫ sin x dx = −cos x + C (NEGATIVE); ∫ cos x dx = sin x + C (POSITIVE); don't mix up signs
Last minute tips
- When finding area between curves, always SET EQUAL and SOLVE for intersection points first. Test a point between intersections to confirm which curve is upper/lower. If roles reverse, SPLIT the integral.
- For centroid ȳ with vertical strips: the integrand is (1/2)f(x)·f(x) = (1/2)[f(x)]² in the numerator. This accounts for the strip's own vertical position. Forgetting the 1/2 is the #1 centroid error—verify by dimensional analysis: M_x should be (length)³ × width = (length)⁴ dimension.
- Disk/washer volume: make sure you identify the AXIS OF REVOLUTION correctly. If about x-axis, integrate dx and use y-values as radii. If about y-axis, integrate dy and use x-values as radii. Draw a quick 3D sketch to avoid axis confusion.
- In definite integrals, ALWAYS evaluate the antiderivative at BOTH bounds: F(upper) − F(lower). Common exam mistake: evaluate at only one bound or reverse the order (wrong sign). Write it as [F(x)]_a^b = F(b) − F(a) explicitly.
- For shell method, the factor 2π is non-negotiable. If revolving about y-axis, use V = 2π ∫ x·f(x) dx (radius = x-distance, height = f(x) = vertical extent). If about x-axis, use V = 2π ∫ y·g(y) dy. Mixing methods wastes time—choose one and commit.
Comparison tables
Disk vs Washer vs Shell Methods
| Method | Setup | When to Use | Formula | Common Axis |
|---|---|---|---|---|
| Disk | Single radius R from axis to curve; stacked perpendicular to axis | Single curve; solid region (no hole) | V = π ∫ R² da | x-axis, y-axis |
| Washer | Two radii: R_outer (top) and R_inner (bottom); stacked perpendicular to axis | Region between two curves; creates annular (ring) cross-section | V = π ∫ (R_outer² − R_inner²) da | x-axis, y-axis |
| Shell | Cylindrical shells with radius r, height h, thickness dr (or dy); summed radially | Any curve; especially useful if axis of revolution not aligned with natural strip orientation | V = 2π ∫ r·h da | y-axis (x ∫), x-axis (y ∫) |
Centroid Formulas: Vertical Strip vs Composite
| Quantity | Vertical Strip (y = f(x)) | Horizontal Strip (x = g(y)) | Composite Shapes |
|---|---|---|---|
| Area Element | dA = f(x) dx | dA = g(y) dy | A = Σ A_i |
| First Moment M_y | M_y = ∫ x·f(x) dx | M_y = ∫ x·g(y) dy | M_y = Σ (x̄_i · A_i) |
| First Moment M_x | M_x = ∫ (1/2)[f(x)]² dx | M_x = ∫ y·g(y) dy | M_x = Σ (ȳ_i · A_i) |
| Centroid x̄ | x̄ = M_y/A | x̄ = M_y/A | x̄ = Σ(x̄_i·A_i) / Σ A_i |
| Centroid ȳ | ȳ = M_x/A | ȳ = M_x/A | ȳ = Σ(ȳ_i·A_i) / Σ A_i |
Integration Methods & Common Pitfalls
| Technique | Best For | Formula / Process | Watch Out |
|---|---|---|---|
| U-Substitution | Composite functions (chain rule reverse) | Let u = g(x), du = g'(x) dx, substitute, integrate ∫ f(u) du, back-substitute | du MUST match integrand; don't forget back-substitution; check definite integral bounds change |
| Integration by Parts | Products: x·sin x, x·e^x, ln x·(poly) | ∫ u dv = uv − ∫ v du; LIATE rule: u = Logarithm, Inverse trig, Algebra, Trig, Exponential (top priority) | Wrong u choice leads to harder integral; may cycle (apply parts twice); watch signs; ∫ v du can be as hard as original |
| Partial Fractions | Rational functions P(x)/Q(x), deg(P) < deg(Q) | Factor Q(x); decompose into A/(x−a) + B/(x−b) + ... ; match coefficients or substitute convenient x values | If deg(P) ≥ deg(Q), do long division first; repeated factors: (x−a)² requires both A/(x−a) and B/(x−a)² |
Standard Antiderivatives At-a-Glance
| Function Type | Antiderivative | Conditions / Notes |
|---|---|---|
| x^n | x^(n+1)/(n+1) + C | n ≠ −1; add 1 to exponent, divide by new exponent |
| 1/x | ln|x| + C | Absolute value essential; domain: x ≠ 0 |
| e^x | e^x + C | No coefficient; e^(kx) → (1/k)e^(kx) + C |
| a^x | a^x / ln(a) + C | a > 0, a ≠ 1; denominator is ln(a), NOT log(a) |
| sin x | −cos x + C | Negative sign; sin(kx) → −(1/k)cos(kx) + C |
| cos x | sin x + C | Positive sign; cos(kx) → (1/k)sin(kx) + C |
| tan x | −ln|cos x| + C OR ln|sec x| + C | Either form equivalent |
| sec^2 x | tan x + C | Inverse of tan; sec^2(kx) → (1/k)tan(kx) + C |
| csc^2 x | −cot x + C | Negative sign; csc^2(kx) → −(1/k)cot(kx) + C |
| sec x tan x | sec x + C | sec(kx)tan(kx) → (1/k)sec(kx) + C |
| 1/√(1−x²) | arcsin x + C | Domain: |x| < 1; ∫ 1/√(a²−x²) dx = arcsin(x/a) + C |
| 1/(1+x²) | arctan x + C | Domain: all x; ∫ 1/(a²+x²) dx = (1/a)arctan(x/a) + C |
Diagrams
Integral Calculus Formula Map
Mind map showing the hierarchical structure of integral calculus topics, from basic antiderivatives through techniques, definite integrals, volumes, and applications. Useful for understanding how all concepts interconnect.
Choosing the Right Integration Method
Flowchart for deciding which integration method to use. Guides students through the decision tree: indefinite vs definite, recognizing standard forms, identifying products vs composites, and applying the appropriate technique (power rule, substitution, by parts, or partial fractions).
Area Between Curves: Step-by-Step Solver
Step-by-step flowchart for solving area between curves problems. Emphasizes finding intersections, determining which is upper/lower via test points, checking for crossings within the interval, and splitting integrals if necessary.
Volume of Revolution: Method Selection
Flowchart for selecting the correct method (disk, washer, or shell) for volume of revolution problems. Based on axis of revolution and shape of region, guides students to the appropriate formula.
Centroid Calculation State Machine
State diagram showing the sequential steps for calculating a centroid: sketch the region, choose strip orientation (vertical or horizontal), compute total area, compute first moments, and divide to get centroid coordinates. Highlights the critical formula for M_x with vertical strips: (1/2)f² (not just f²).