Before You Start
For every area-between-curves problem, your first job is not to integrate. Your first job is to understand the region.
If you integrate with respect to , think in terms of top minus bottom:
If you integrate with respect to , think in terms of right minus left:
When the curves intersect, you will usually need to find the intersection points before writing the integral. Also check whether the curves switch which one is on top or on the right inside the region.
For each problem, write down the bounds, the top/bottom or right/left functions, and the integral before evaluating. On an AP free-response question, the setup is often as important as the final numerical answer.
Practice Problems
Find the area of the region bounded by and from to .
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Find the area enclosed by and .
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Find the area of the region enclosed by and .
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Find the area of the region bounded by and , using integration with respect to .
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Find the total area between and on .
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Find the area between the curves and on .
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Let and . Set up, but do not evaluate, an integral representing the area of the region enclosed by the two curves and the -axis.
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Find the total area enclosed by and .
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How to Check Your Setup
Before pressing integrate, ask yourself four questions:
- Did I find the correct intersection points?
- Am I integrating in the easier direction? Vertical slices use top minus bottom; horizontal slices use right minus left.
- Is the function I subtracted actually below or to the left throughout the interval?
- Does the resulting area have to be split into multiple integrals? If the curves cross inside your interval, the answer may be yes.
A definite integral can be negative, but geometric area cannot. If your result is negative, check whether you accidentally used bottom minus top or left minus right. If the curves cross, split the integral at the crossing point rather than allowing positive and negative signed areas to cancel.
Quick Reference
The most important skill is choosing the correct setup. Once the bounds and integrand are correct, the integration itself is usually the easy part.
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