Integration and Areas Under Curves Explained
Integration is the second fundamental operation of calculus, complementing differentiation. Where differentiation finds rates of change, integration accumulates quantities — finding areas, volumes, total distances, and total changes over an interval. Together these two operations are linked by one of the most beautiful results in all of mathematics.
Antiderivatives and Indefinite Integrals
If differentiation asks "what is the rate of change of f(x)?", integration asks the reverse: "which function F(x) has f(x) as its derivative?" F(x) is called an antiderivative of f(x). Because the derivative of any constant is zero, antiderivatives are not unique — if F(x) is an antiderivative of f(x), so is F(x) + C for any constant C. We write the indefinite integral as:
∫ f(x) dx = F(x) + C
The integral sign ∫ is an elongated S (for "sum"), and dx indicates that x is the variable of integration. The constant of integration C is always included for an indefinite integral; its value is determined when additional information (such as a boundary condition) is given.
Standard Integral Rules
The power rule for integration reverses the power rule for differentiation — increase the exponent by one and divide by the new exponent:
∫ xn dx = xn+1 / (n+1) + C (provided n ≠ −1)
The special case n = −1 gives ∫ (1/x) dx = ln|x| + C.
| f(x) | ∫ f(x) dx |
|---|---|
| xn (n ≠ −1) | xn+1/(n+1) + C |
| 1/x | ln|x| + C |
| ex | ex + C |
| ekx | (1/k)ekx + C |
| sin(x) | −cos(x) + C |
| cos(x) | sin(x) + C |
| sec2(x) | tan(x) + C |
Definite Integrals
A definite integral has limits: it integrates from a lower bound a to an upper bound b, and the result is a specific number rather than a family of functions.
∫ab f(x) dx = F(b) − F(a)
The constant of integration cancels out in a definite integral, so it is not written. The result gives the signed area between the curve y = f(x) and the x-axis from x = a to x = b: area above the x-axis contributes positive values, and area below contributes negative values.
The Fundamental Theorem of Calculus
The relationship between differentiation and integration is formalised in the Fundamental Theorem of Calculus, proved independently by Newton and Leibniz in the 17th century. It has two parts:
- If F(x) = ∫ax f(t) dt, then F'(x) = f(x). In other words, differentiating an integral with a variable upper limit gives back the original function — differentiation and integration are inverse operations.
- ∫ab f(x) dx = F(b) − F(a), where F is any antiderivative of f. This is the rule used in all practical calculations.
This theorem bridges the geometric idea of area (accumulated infinitesimally thin rectangles) with the algebraic idea of finding antiderivatives — a profound and non-obvious connection.
Finding Areas Under Curves: Worked Example
Find the area between the curve y = x2 and the x-axis from x = 1 to x = 4.
Step 1: Write the definite integral: ∫14 x2 dx.
Step 2: Find the antiderivative: F(x) = x3/3.
Step 3: Evaluate: F(4) − F(1) = 64/3 − 1/3 = 63/3 = 21 square units.
Area Between Two Curves
When the area between two curves y = f(x) and y = g(x) is required, first find where they intersect (solving f(x) = g(x) for x), then integrate the difference of the upper and lower functions between those limits:
Area = ∫ab [f(x) − g(x)] dx (where f(x) ≥ g(x) on [a, b])
Always sketch the curves first to identify which is the upper function across the interval. If the curves cross within the interval, split the integral at the crossing point.
Integration Techniques Beyond the Basics
Simple integrals are handled by standard rules, but more complex integrands require further techniques:
- Integration by substitution (reverse chain rule): replace a composite expression with a single variable u, integrate, then substitute back. Example: ∫ 2x(x2+1)4 dx. Let u = x2+1, du = 2x dx, so the integral becomes ∫ u4 du = u5/5 + C = (x2+1)5/5 + C.
- Integration by parts (reverse product rule): ∫ u dv = uv − ∫ v du. Choose u and dv using the LIATE priority order (Logarithm, Inverse trig, Algebraic, Trig, Exponential).
- Partial fractions: decompose a rational function into simpler fractions before integrating.
Summary
Integration reverses differentiation: the indefinite integral ∫ f(x) dx = F(x) + C finds the antiderivative family, while the definite integral ∫ab f(x) dx = F(b) − F(a) computes a specific signed area. The Fundamental Theorem of Calculus unifies these ideas. Standard rules (power, exponential, trigonometric) cover most cases; substitution, integration by parts, and partial fractions handle more complex integrands. Together with differentiation, integration forms the core of calculus and underpins physics, engineering, probability, and economics.