CELE Engineering Mathematics — Integral CalculusRevision Notes

Formulas

Exam Tips

• Always verify your antiderivative by differentiating your answer — this takes 10 seconds and confirms correctness.
• When you see a composite function (e.g., (2x+3)⁵), immediately attempt u-substitution with u = inner function.
• For integrals of the form ∫sin²x dx or ∫cos²x dx, apply the half-angle identity first before integrating.
• Memorize the five standard integrals (power, ln, eˣ, sin, cos) perfectly — they appear as sub-steps in every application problem.
• In multiple-choice format, substitute x = 1 (or another simple value) into both your answer and the answer choices to eliminate wrong options quickly.

Key Points

• Integration reverses differentiation: if d/dx[F(x)] = f(x), then ∫f(x)dx = F(x) + C, where C is the constant of integration — ALWAYS include C for indefinite integrals.
• The Power Rule ∫xⁿdx = xⁿ⁺¹/(n+1) + C is valid for all n ≠ −1; the exception ∫dx/x = ln|x| + C must be memorized separately.
• Standard trigonometric integrals: ∫sin x dx = −cos x + C and ∫cos x dx = sin x + C; note the sign change for sine.
• Exponential integral: ∫eˣdx = eˣ + C; for ∫eᵃˣdx = (1/a)eᵃˣ + C.
• Substitution (u-substitution) is the most tested technique: let u = inner function, compute du, convert everything to u, integrate, then back-substitute.
• Integration by Parts formula: ∫u dv = uv − ∫v du. Use the LIATE priority rule to select u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.
• Partial Fractions: decompose a rational function P(x)/Q(x) (degree of P < degree of Q) into simpler fractions before integrating.
• Trigonometric identities frequently simplify integrals: sin²x = (1−cos2x)/2 and cos²x = (1+cos2x)/2 are essential.

Definitions

Section Title

Basic Integration Rules and Techniques

Common Mistakes

• Omitting the constant of integration C in indefinite integrals — automatic point deduction.
• Applying the power rule to n = −1: ∫x⁻¹dx ≠ x⁰/0 — this is undefined; the correct answer is ln|x| + C.
• Forgetting to back-substitute after u-substitution: leaving the answer in terms of u instead of the original variable x.
• Sign error in ∫sin x dx: the result is −cos x + C, not +cos x + C.
• In integration by parts, choosing u as the exponential function when an algebraic or logarithmic term is present — violates LIATE and complicates the integral.
• Not fully converting the integral to u after substitution: mixing x and u terms in the integrand.

Formulas

Exam Tips

• Always sketch a rough graph of both curves and mark intersection points before setting up the integral — 30 seconds of sketching prevents major errors.
• Check: if the computed area is negative, you reversed the upper and lower curves — swap and re-evaluate.
• For parabola-vs-line problems (extremely common in board exams), the intersection points are the limits; area = ∫(line − parabola)dx in most cases.
• When limits are given (e.g., 0 to 2), verify which curve is above the other by testing a midpoint x-value.
• Memorize the result: area under y = 4 − x² above x-axis = ∫₋₂² (4−x²)dx = 32/3 ≈ 10.67 — this specific problem type recurs in Philippine board exams.

Key Points

• The Fundamental Theorem of Calculus: ∫ₐᵇ f(x)dx = F(b) − F(a), where F is any antiderivative of f. No constant C needed.
• Geometrically, the definite integral gives the net signed area between f(x) and the x-axis from x = a to x = b.
• Area is always positive: if f(x) < 0 on [a,b], take the absolute value or integrate |f(x)|.
• Area between two curves (upper minus lower): A = ∫ₐᵇ [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a,b].
• Find intersection points first by solving f(x) = g(x) — these become the limits of integration when not given.
• If the curves switch positions (one becomes upper, then lower), split the integral at the crossing point and add absolute values.
• Horizontal strip method: sometimes integrating with respect to y (with rightmost minus leftmost functions expressed in x = h(y)) is simpler for sideways parabolas or regions bounded by y-axis.
• Properties: ∫ₐᵇ f dx = −∫ᵦᵃ f dx; ∫ₐᵃ f dx = 0; ∫ₐᶜ f dx = ∫ₐᵇ f dx + ∫ᵦᶜ f dx.

Definitions

Section Title

Definite Integrals and Area Between Curves

Common Mistakes

• Not finding intersection points first — using arbitrary or wrong limits of integration produces a completely incorrect area.
• Subtracting in the wrong order (lower minus upper instead of upper minus upper) — gives a negative area.
• Failing to split the integral when curves cross within the interval — underestimates the true enclosed area.
• Applying the Fundamental Theorem without computing F(b) − F(a) correctly — arithmetic errors in substituting limits.
• Forgetting to set g(x) = 0 when the lower curve is the x-axis — treat it explicitly as y = 0.

Formulas

Exam Tips

• Always write out the radius R(x) explicitly before squaring — this one step prevents the [R−r]² mistake.
• Quick check: disk volume about x-axis for y = f(x) from a to b must be π times the integral of f(x)² — if your result lacks π, something is wrong.
• For revolution about the y-axis, compare effort: shell method integrates directly in x; disk method requires solving for x = g(y) and re-expressing limits in y. Shell is usually faster.
• When the axis is NOT the x-axis or y-axis (e.g., y = 2), the radius becomes R(x) = 2 − f(x) (if below y = 2) or f(x) − 2 (if above). Redraw the axis explicitly.
• Memorize: volume of sphere = (4/3)πr³ — you can derive this by revolving a semicircle y = √(r²−x²) from −r to r using the disk method. If your formula gives this, your setup is correct.

Key Points

• A solid of revolution is formed by rotating a plane region about a line (axis of revolution).
• Disk Method (solid of revolution, about x-axis): V = π∫ₐᵇ [R(x)]² dx, where R(x) is the radius (distance from curve to axis).
• Washer Method (hollow solid, about x-axis): V = π∫ₐᵇ {[R(x)]² − [r(x)]²} dx, where R = outer radius, r = inner radius.
• Shell Method (about y-axis): V = 2π∫ₐᵇ x·f(x) dx. The integrand is (shell radius)×(shell height) = x·f(x).
• For revolution about y-axis using disk/washer, express x as a function of y and integrate with respect to y.
• For revolution about x-axis using shell, express y as a function of y (shell height is in y), integrate w.r.t. y.
• Choosing disk vs. shell: disk is natural when revolving about the x-axis with vertical strips; shell is natural when revolving about the y-axis with vertical strips — no need to re-express the function in terms of y.
• Always confirm the axis of revolution and whether the region generates a solid (disk) or hollow (washer) cross-section.

Definitions

Section Title

Volumes of Revolution

Common Mistakes

• Forgetting to square the radius in the disk/washer method — integrating R(x) instead of [R(x)]².
• In the washer method, squaring the difference instead of the difference of squares: [R − r]² ≠ R² − r².
• Using the shell method formula 2π∫x·f(x)dx but integrating with respect to y instead of x.
• Not identifying which curve is outer vs. inner in the washer method — setting up R and r incorrectly.
• Revolving about a shifted axis (e.g., x = 3 or y = −1) without adjusting the radius expression: R(x) = (3 − x) or R(y) = (y + 1), not simply x or y.
• Applying the disk formula when a washer is needed — ignoring the hollow region produces an overestimate.

Formulas

Exam Tips

• Always compute A first, then Mᵧ and Mₓ separately — never skip the area computation even if it seems obvious.
• The formula ȳ = [∫(1/2)(f(x))²dx] / [∫f(x)dx] is a direct shortcut — memorize it exactly in this form to avoid the y vs. y/2 error.
• Pappus's Theorem is your best cross-check: after computing a volume by disk/shell, verify using V = 2π·x̄·A if you know the centroid.
• For composite areas (T-beams, L-shapes), use the weighted average formula x̄ = ΣAᵢx̄ᵢ / ΣAᵢ — this is the discrete analogue of the integral centroid.
• If the problem gives a region between two curves, the total area A = ∫(f−g)dx, and Mᵧ = ∫x(f−g)dx, Mₓ = ∫(1/2)(f²−g²)dx.

Key Points

• The centroid (x̄, ȳ) is the geometric center of a plane area, equivalent to the center of gravity for a uniform density lamina.
• Total area: A = ∫ dA = ∫ₐᵇ f(x) dx (for area under a single curve above the x-axis).
• Moment about y-axis (first moment): Mᵧ = ∫ x dA = ∫ₐᵇ x·f(x) dx → centroid x̄ = Mᵧ/A.
• Moment about x-axis: Mₓ = ∫ y_el dA where y_el = f(x)/2 (centroid of the vertical strip) → ȳ = Mₓ/A.
• Critical insight: the centroid of a vertical strip of height y = f(x) is at y/2, NOT y. Using y instead of y/2 for Mₓ is the most common centroid error.
• Moment of Inertia (second moment of area): Iₓ = ∫ y² dA = ∫ₐᵇ [f(x)]³/3 dx (for vertical strip about x-axis).
• Arc length: L = ∫ₐᵇ √(1 + [f'(x)]²) dx — integration of the differential arc length element ds.
• Pappus's Theorem (shortcut): Volume = 2π·ȳ·A (revolution about x-axis) or V = 2π·x̄·A (about y-axis). Useful for verification.

Definitions

Section Title

Centroids and Moments by Integration

Common Mistakes

• Using y instead of y/2 as the y-centroid of a vertical strip: the strip's own centroid is at its midpoint y/2, so Mₓ = ∫(y/2)(y dx) = ∫(1/2)y² dx, NOT ∫y·y dx = ∫y² dx.
• Computing ȳ = ∫y dA / A using the wrong expression for y — students sometimes plug in f(x) directly without the 1/2 factor.
• In arc length, forgetting to differentiate f(x) before squaring — using f(x)² instead of [f'(x)]² inside the square root.
• Applying Pappus's Theorem for revolution about the x-axis using x̄ instead of ȳ — must use the centroid coordinate perpendicular to the axis.
• Setting up moment integrals without a common denominator of A — computing numerator alone and reporting it as the centroid.