CELE Engineering Mathematics — Integral CalculusSummary

An antiderivative F(x) of a function f(x) satisfies F'(x) = f(x). The indefinite integral ∫f(x)dx represents the family of all antiderivatives and is written as F(x) + C, where C is the constant of integration. Basic rules include: ∫x^n dx = x^(n+1)/(n+1) + C (for n ≠ -1), ∫(1/x)dx = ln|x| + C, ∫e^x dx = e^x + C, ∫sin(x)dx = -cos(x) + C, and ∫cos(x)dx = sin(x) + C. The constant C represents an infinite family of vertical shifts of the same function — a critical detail on PRC exams.

The definite integral ∫[a,b] f(x)dx = F(b) − F(a), where F is any antiderivative of f. This represents the net accumulation of f(x) over the interval [a, b]. The Fundamental Theorem connects differentiation and integration: if F'(x) = f(x), then ∫[a,b] f(x)dx = F(b) − F(a). Geometrically, a definite integral equals the signed area under the curve — positive above the x-axis, negative below. The limits a (lower) and b (upper) are crucial; reversing them negates the result.

When two curves y = f(x) (upper) and y = g(x) (lower) intersect at x = a and x = b, the area enclosed is A = ∫[a,b] [f(x) − g(x)]dx. Critical steps: (1) identify which curve is above the other in the interval, (2) find intersection points algebraically, (3) set up the integral with upper minus lower, (4) integrate and evaluate. If curves cross within the interval, split the integral at crossing points. For vertical strips (integrating with respect to x), the height of each strip is the difference of the y-values. For horizontal strips (integrating with respect to y), solve for x and use the difference of x-values.

When a planar region is revolved about an axis, it generates a solid of revolution. The disk method (revolving about the x-axis) uses V = π∫[a,b] [R(x)]² dx, where R(x) is the distance from the axis to the curve (the radius). The washer method extends this: V = π∫[a,b] ([R_outer(x)]² − [R_inner(x)]²) dx, subtracting the volume of the inner hollow region. When revolving about the y-axis, use R as the horizontal distance. The key is identifying the outer radius (farther from axis) and inner radius (closer to axis, if any). Common error: forgetting to square the radius or confusing which function is outer/inner.

For solids generated by revolving about the y-axis, the shell method is often simpler: V = 2π∫[a,b] x·f(x) dx. Here, x is the radius of the cylindrical shell (distance from the y-axis), and f(x) is the height of the shell. A vertical strip at position x (thickness dx) at height f(x) traces a thin cylindrical shell when revolved. The factor 2πx is the circumference of the shell, f(x) is its height, and dx is its thickness. Choose this method when the solid is easier to describe using vertical strips and revolution about a vertical axis.

Substitution (u-substitution) simplifies complex integrals. Steps: (1) identify a substitution u = g(x) such that du/dx = g'(x), so du = g'(x)dx, (2) rewrite the integral in terms of u, (3) integrate with respect to u, (4) substitute back to x. Example: ∫2x·e^(x²)dx — let u = x², then du = 2x dx, so the integral becomes ∫e^u du = e^u + C = e^(x²) + C. For definite integrals, change the limits: if u = g(x), then new limits are u(a) and u(b). Critical error: forgetting to substitute both the integrand and the differential (dx).

For products of functions, use ∫u dv = uv − ∫v du. Strategy: choose u as the function that becomes simpler when differentiated (often logarithmic or polynomial), and dv as the remaining part. Example: ∫x·e^x dx — let u = x (so du = dx) and dv = e^x dx (so v = e^x). Then ∫x·e^x dx = x·e^x − ∫e^x dx = x·e^x − e^x + C = e^x(x − 1) + C. LIATE rule helps select u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential — prioritize earlier categories as u. Integration by parts may need to be applied repeatedly.

Decompose rational functions (polynomials over polynomials) into simpler fractions before integrating. For a proper fraction (degree of numerator < degree of denominator), factor the denominator and express the fraction as a sum of partial fractions. Example: (1/(x² − 1)) = (1/((x−1)(x+1))) = (1/2)·(1/(x−1)) − (1/2)·(1/(x+1)). Then ∫(1/(x² − 1))dx = (1/2)ln|x−1| − (1/2)ln|x+1| + C. For improper fractions, perform polynomial long division first. Each factor (x − a) in the denominator contributes a term A/(x − a); repeated factors (x − a)^n contribute multiple terms.

The centroid (geometric center) of a planar region is found by: x̄ = ∫x dA / ∫dA and ȳ = ∫y_centroid dA / ∫dA, where y_centroid is the y-coordinate of the centroid of each differential area element. For a vertical strip of width dx and height y, dA = y·dx and the centroid of the strip is at (x, y/2). Thus ȳ = ∫(y/2)·(y dx) / ∫(y dx) = ∫(y²/2) dx / ∫y dx. A common error is using y instead of y/2 in the numerator. Centroids are essential for structural analysis — the centroid of cross-sections determines neutral axes for bending.

The second moment of area (often called moment of inertia in engineering contexts, denoted I) measures resistance to bending: I_x = ∫y² dA and I_y = ∫x² dA. For a vertical strip, dA = y dx, so I_x = ∫y² · (y dx) = ∫y³ dx. The polar moment J = I_x + I_y. The parallel axis theorem relates moments about different axes: I = I_centroid + A·d², where d is the perpendicular distance between axes and A is the total area. These are critical for designing beams (AISC 360) and calculating deflections.

The length of a curve y = f(x) from x = a to x = b is given by L = ∫[a,b] √(1 + (dy/dx)²) dx. For curves given as x = g(y), use L = ∫[c,d] √(1 + (dx/dy)²) dy. The integrand √(1 + (dy/dx)²) represents the length of an infinitesimal segment. Parametric curves use L = ∫√((dx/dt)² + (dy/dt)²) dt. Arc length integrals are often difficult to evaluate analytically; numerical methods are sometimes used in practice.

Integration is the inverse of differentiation. If F'(x) = f(x), then ∫f(x)dx = F(x) + C. The Fundamental Theorem formalizes this: ∫[a,b] f(x)dx = F(b) − F(a). Understanding this inverse relationship is key to applying integration in solving real-world problems.

A definite integral geometrically represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b. When f(x) > 0, area is positive; when f(x) < 0, area is negative (below the axis). This concept extends to areas between two curves (upper minus lower) and forms the basis for volume calculations.

When an area between curves is revolved about an axis, the disk/washer method generalizes the area concept to three dimensions. The cross-sectional area A(x) = π[R(x)]² is integrated along the axis of revolution to yield volume. This hierarchical relationship shows how 2D integrals extend to 3D.

Substitution works for composite functions (chain rule inverse), integration by parts for products (product rule inverse), and partial fractions for rational functions. Recognizing which method applies requires pattern recognition—a skill developed through practice on varied PRC-style problems.

The centroid of a cross-section determines its neutral axis (the line through which bending moments cause no stress). NSCP 2015 and AISC 360 design procedures rely on centroid calculations to determine bending stresses σ = M·y/I, where y is distance from the neutral axis and I is the second moment of inertia.

First moments ∫x dA and ∫y dA yield centroid coordinates when divided by area A. Second moments (moments of inertia) ∫x² dA and ∫y² dA quantify resistance to bending and are used in deflection and stress calculations. Both are essential for structural design per AISC 360.

In structural analysis per NSCP 2015, distributed loads are represented as continuous functions. The total load on a beam segment is Q = ∫[x₁,x₂] w(x) dx, where w(x) is load per unit length. Shear force V(x) = ∫w(x) dx + C, and bending moment M(x) = ∫V(x) dx + C. Understanding integration is essential for constructing shear and moment diagrams.

Steel I-beams and concrete sections are analyzed by dividing them into simple shapes (rectangles, triangles), calculating the centroid and I for each part, then using ȳ = ΣA·ȳ_i / ΣA and the parallel axis theorem I = ΣI_centroid + ΣA·d². Integration extends this to curved boundaries (circular sections, fillets).

Hydrostatic pressure at depth h is p(h) = ρgh (where ρ is fluid density, g is gravity). The total force on a submerged vertical surface of height H is F = ∫[0,H] p(h)·A(h) dh, where A(h) is the horizontal width at depth h. For a rectangular surface, F = ρg·∫[0,H] h·b dh = ρg·b·H²/2.

For cylindrical tanks revolved about a vertical axis, V = π∫[0,H] [R(h)]² dh. For conical or more complex shapes, the cross-sectional area A(h) varies with height. Integration determines tank capacity and is essential for sizing municipal water systems per Philippine design standards.

The differential equation for beam deflection is EI(d⁴y/dx⁴) = M(x), where E is elastic modulus, I is moment of inertia, y is deflection, and M(x) is bending moment. Solving requires integrating four times to obtain y(x). Double integration (∫∫M(x)dx²) gives the deflection curve, which must satisfy boundary conditions (support conditions).

For retaining walls, dams, and floating structures, stability depends on the location of the center of gravity (centroid). Overturning safety factor = M_resisting / M_overturning requires accurate centroid calculations using integration for non-rectangular sections.

For shafts and members under torsion, shear stress τ = T·r/J, where T is torque, r is radial distance, and J is polar moment of inertia. For non-circular sections (e.g., rectangular or complex profiles), J = ∫(x² + y²) dA requires integration to determine.

The catenary curve y = a·cosh(x/a) models cables under their own weight. Arc length L = ∫√(1 + (dy/dx)²) dx determines cable length and sag in suspension bridges and power transmission lines.