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CALCULUS · INTEGRALS

The Fundamental Theorem

Integration and differentiation are inverse operations — and that makes area easy.

16 min read 4 practice questions
An antiderivative of ff is a function FF with F′=fF' = f. The Fundamental Theorem of Calculus says the exact area under ff is the change in any antiderivative.
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)
∫xn dx=xn+1n+1+C(n≠−1)\int x^n\,dx = \frac{x^{n+1}}{n + 1} + C \quad (n \neq -1)

Reverse power rule

Worked example · Exact area

Evaluate ∫04x2 dx\int_0^4 x^2\,dx.
  1. 1
    Antiderivative: F(x)=x33F(x) = \frac{x^3}{3}.
  2. 2
    F(4)−F(0)=643−0=643≈21.33F(4) - F(0) = \frac{64}{3} - 0 = \frac{64}{3} \approx 21.33.

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

1
Evaluate ∫132x dx\int_1^3 2x\,dx.
2
Find ∫(6x2+4) dx\int (6x^2 + 4)\,dx.
3
Evaluate ∫02(3x2+1) dx\int_0^2 (3x^2 + 1)\,dx.
4
If G(x)=∫0xcos⁡(t2) dtG(x) = \int_0^x \cos(t^2)\,dt, what is G′(x)G'(x)?

Numeric answers accept decimals or fractions like 3/4.