01 Key Concepts
Definite Integral Notation
Integral from a to b of f(x) dx represents the net signed area under f(x) between x=a and x=b.
Fundamental Theorem of Calculus
If F is an antiderivative of f, then Integral from a to b of f(x) dx = F(b) - F(a).
Signed Area
Area above the x-axis counts as positive; area below counts as negative.
Properties of Definite Integrals
Integral from a to a = 0. Integral from a to b = -Integral from b to a. Integrals over adjoining intervals add together.
No '+C' Needed
Since we evaluate F(b) - F(a), any constant C cancels out, so definite integrals give one exact number.
02 Key Formulas
- Integral from a to b of f(x) dx = F(b) - F(a)
03 Solved Examples
Example 1 Evaluate the integral from 0 to 2 of x^2 dx.
- Find the antiderivative: F(x) = x^3/3.
- Evaluate at the bounds: F(2) - F(0) = 8/3 - 0.
Answer: 8/3
Example 2 Evaluate the integral from 1 to 3 of (2x + 1) dx.
- Find the antiderivative: F(x) = x^2 + x.
- Evaluate: F(3) - F(1) = (9+3) - (1+1) = 12 - 2.
Answer: 10
Example 3 Evaluate the integral from -1 to 1 of x^3 dx.
- Find the antiderivative: F(x) = x^4/4.
- Evaluate: F(1) - F(-1) = 1/4 - 1/4.
Answer: 0 (the positive and negative areas cancel out)
04 Practice Questions
1Evaluate the integral from 0 to 1 of x^2 dx.
1/3
2Evaluate the integral from 0 to 3 of 2 dx.
6
3Evaluate the integral from 1 to 2 of x dx.
3/2
4Evaluate the integral from 0 to 2 of (3x^2) dx.
8
5If Integral from 0 to 5 of f(x)dx = 10, what is Integral from 5 to 0 of f(x)dx?
-10
📄 Definite Integrals — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.