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Integral Calculus

Areas, volumes, accumulation, and the inverse of differentiation — measuring continuous change.

The Definite Integral: Area Under a Curve

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The definite integral of a function f from a to b is the signed area between the graph of f and the x-axis, from x equals a to x equals b. "Signed" means that area above the x-axis counts as positive and area below counts as negative. The notation was invented by Leibniz: an elongated S (for "sum") with limits a and b, followed by f of x dx. The dx indicates that the variable of integration is x and that we are summing infinitesimally thin rectangles. The Fundamental Theorem (FTC) of Calculus (FTC) reveals the profound connection between integration and differentiation: integration undoes differentiation. If F is any antiderivative of f (meaning the derivative of F is f), then the definite integral from a to b of f of x dx equals F of b minus F of a.
The Fundamental Theorem of Calculus. Integration and differentiation are inverse operations. This is one of the deepest theorems in mathematics.

Techniques of Integration

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Computing integrals requires a toolbox of techniques, each suited to different forms. The power rule reverses differentiation of powers. U-substitution is the reverse of the chain rule — set u equal to a function inside the integrand, replace dx with du, and integrate with respect to u. Integration by parts reverses the product rule of differentiation: the integral of u dv equals u times v minus the integral of v du. The mnemonic LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) helps choose u. Trigonometric substitution handles square roots of quadratic expressions by substituting trig functions. Partial fraction decomposition breaks rational functions into simpler fractions that are easier to integrate.
Integration by parts. Choose u (to differentiate) and dv (to integrate) wisely. LIATE helps select u.
Power rule for integration. Add one to the exponent, then divide by the new exponent. The +C represents the constant of integration.

Applications of Integration

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Integration powers nearly every quantitative science. It computes the area between two curves by integrating the difference of their functions. Volumes of solids of revolution are computed by the disk method (integrating cross-sectional circles) or the shell method (integrating cylindrical layers). Arc length of a curve is computed by integrating the square root of one plus the derivative squared. Surface area of a rotated curve follows a similar formula. In physics, integration computes work (the integral of force with respect to displacement), center of mass, moment of inertia, and electric potential. In probability, the integral of a probability density function gives cumulative probability.

Key Integration Techniques

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  • Power rule: for polynomials. Add 1 to exponent, divide by new exponent.
  • U-substitution: reverse chain rule. Replace inner function with u.
  • Integration by parts: reverse product rule. Use LIATE to choose u.
  • Trig substitution: for square root expressions. Use sin, cos, or tan substitutions.
  • Partial fractions: decompose rational functions into a sum of simpler fractions.
  • Numerical integration: Simpson's rule, trapezoidal rule. For when antiderivatives don't exist in closed form.

Worked Examples

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Worked Examples
Example 1: Integrate x² from 0 to 2.
Antiderivative: x³/3. Evaluate: F(2)−F(0) = 8/3 − 0 = 8/3 ≈ 2.667.

Example 2: Integrate 2x×cos(x²) dx by u-substitution.
Let u = x², du = 2x dx. Integral becomes ∫cos(u)du = sin(u) + C = sin(x²) + C.

Example 3: Integrate x×eˣ dx by parts.
Let u = x, dv = eˣ dx. Then du = dx, v = eˣ. So ∫xeˣdx = xeˣ − ∫eˣdx = xeˣ − eˣ + C.

Integral Visualizer

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See the area under a curve change as the upper limit moves. This is the visual meaning of the Fundamental Theorem of Calculus.