Short on time? This page is the recap. For explanations, the graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) f(x) dx is the definite integral of f from a to b.
Recap
- If the region is made of rectangles, triangles, trapezoids or parts of circles, the integral is the sum of their areas, with minus for parts below the axis.
- y = √(r² − x²) is a semicircle of radius r above the axis; its area is (1/2)πr².
- The properties below let you combine integral values you are given, without knowing f.
- Holes and jumps are allowed. Changing f at one point does not change the integral. At a jump, split the interval and use the right formula on each piece.
- Vertical asymptotes are not allowed here. Those are improper integrals, a Calculus BC topic.
Key relationships
| Property | Rule |
|---|---|
| Zero width | ∫ (a to a) f = 0 |
| Reversed limits | ∫ (b to a) f = −∫ (a to b) f |
| Constant multiple | ∫ (a to b) k·f = k ∫ (a to b) f |
| Sum and difference | ∫ (a to b) (f ± g) = ∫ (a to b) f ± ∫ (a to b) g |
| Adjacent intervals | ∫ (a to b) f + ∫ (b to c) f = ∫ (a to c) f (any order of a, b, c) |
| Constant integrand | ∫ (a to b) k dx = k(b − a) |
Not rules: ∫ fg ≠ (∫ f)(∫ g); ∫ |f| ≠ |∫ f| unless f keeps one sign; ∫ (f + k) ≠ ∫ f + k.
Assumptions behind the method
- f and g are integrable on an interval containing all the limits used (continuous, or bounded with finitely many holes or jumps).
- Signed area: a region below the axis gives a negative contribution.
Mistakes to avoid
- Treating an integral as total area and ignoring negative regions.
- Adding k instead of k(b − a) when a constant is added to the integrand.
- Forgetting the minus sign when the limits are reversed.
- Using the formula from the wrong piece at a jump.
- Using r² as the radius in √(r² − x²).
- Splitting a product of functions.
Quick self-check
- ∫ (1 to 9) f = 4 and ∫ (5 to 9) f = 6. Find ∫ (1 to 5) f. (−2)
- Find ∫ (−6 to 6) √(36 − x²) dx. (18π)
- ∫ (2 to 5) f = 7. Find ∫ (5 to 2) [f(x) + 1] dx. (−(7 + 3) = −10)
Next: practice questions.