The Integral

Subtopic: Integration

The integral accumulates quantities—area under curves, total distance from velocity, mass from density. It's the reverse of differentiation. This page covers the definite integral as a limit of sums, the indefinite integral as antiderivative, and the fundamental connection between them.

Introduction

If derivatives answer "how fast is it changing?", integrals answer "how much has accumulated?" Given a velocity function, the integral recovers total distance. Given a rate of flow, the integral gives total volume.

Geometrically, the definite integral computes the (signed) area between a curve and the x-axis.

The Definite Integral

The definite integral of ƒ from a to b is

(∫_a^b)(ƒ*d(x))=(lim_)((∑_i=1^n)(ƒ((x_i)∗)))*Δ(x)

where Δ(x)=(b−a)/n and (x_i)∗ is a sample point in the i-th subinterval.

This is a Riemann sum: approximate the area using rectangles, then take the limit as the rectangles become infinitely thin.

Geometric Interpretation

For ƒ(x)≥0, the integral (∫_a^b)(ƒ(x)*d(x)) equals the area under the curve y=ƒ(x) from x=a to x=b.

If ƒ can be negative, the integral gives signed area: positive above the x-axis and negative below.

Total area (ignoring sign) requires (∫_a^b)(|ƒ((x_))|d(x)).

The Indefinite Integral

The indefinite integral (antiderivative) of ƒ is a function F such that F′=ƒ:

(∫_^)(ƒ((x_))*d(x))=F(x)+C

The +C matters: if F′=ƒ, then (F+C)′=ƒ too. Antiderivatives are unique only up to a constant.

Fundamental Theorem of Calculus

The key connection between derivatives and integrals:

Part 1

If F(x)=(∫_a^x)(ƒ(t)*d(t)), then F′*(x)=ƒ(x).

Differentiating an integral with respect to its upper limit gives back the integrand.

Part 2

(∫_a^b)(ƒ((x_))*d(x))=F(b)−F(a)

where F is any antiderivative of ƒ. To evaluate a definite integral: find an antiderivative, plug in the limits, subtract.

Worked Example

Compute (∫_1^3)(x2*d(x)).

Step 1: Find the antiderivative of x2. Since (x3)′=3*x2, we have (∫_^)(x2*d(x))=(x3)/3+C.

Step 2: Apply FTC Part 2:

(∫_1^3)(x2*d(x))=[(x3)/3]31=27/3−1/3=9−1/3=26/3

This is the area under the parabola y=x2 from x=1 to x=3.

Basic Integration Rules

Power Rule

(∫_^)(xn*d(x))=(x(n+1))/(n+1)+C*(n≠−1)

Special Case

(∫_^)(1/x*d(x))=ln(|)*x|+C

Exponential

(∫_^)(ex*d(x))=ex+C

Trigonometric

(∫_^)(sin(x)*d(x))=−cos(x)+C

(∫_^)(cos(x)*d(x))=sin(x)+C

Properties

Linearity: ∫[a*ƒ(x)+b*g(x)]*d(x)=a*(∫_^)(ƒ(x)*d(x))+b*(∫_^)(g(x)*d(x))

Additivity: (∫_a^b)(ƒ(x)*d(x))+(∫_b^c)(ƒ(x)*d(x))=(∫_a^c)(ƒ(x)*d(x))

Reversal: (∫_a^b)(ƒ(x)*d(x))=−(∫_b^a)(ƒ(x)*d(x))

Comparison: If ƒ(x)≤g(x) on [a,b], then (∫_a^b)(ƒ(x)*d(x))≤(∫_a^b)(g(x)*d(x))

Applications

In physics, integrating velocity gives displacement; integrating acceleration gives velocity.

In geometry, integrals compute areas, volumes, arc lengths, and surface areas.

In probability, the integral of a density function over an interval gives probability.

In economics, integrating marginal cost gives total cost.

Summary

The definite integral is the limit of Riemann sums—it computes signed area under a curve. The indefinite integral finds antiderivatives. The Fundamental Theorem of Calculus states that if F′=ƒ, then(∫_a^b)(ƒ(x)*d(x))=F(b)−F(a).

Key rules include the power rule, exponential functions, and trigonometric functions. Integration is linear.