Area Under a Curve From an Antiderivative

Area from an antiderivative

When a problem gives you an antiderivative, use the Fundamental Theorem of Calculus to find the area on an interval. If the curve stays above the x-axis, the area from (a) to (b) is [ A = F(b)-F(a), ] where (F) is an antiderivative of the function.

Method

1. Identify the interval: read the lower and upper limits carefully.
2. Use the antiderivative: evaluate the given antiderivative at both endpoints.
3. Subtract in the correct order: compute (F(b)-F(a)), not the reverse.
4. Simplify completely: reduce fractions, combine terms, and write the final exact answer.

Important check

Make sure the answer is an area, so it should not be negative. If the function can change sign on the interval, then (F(b)-F(a)) gives signed area, not always geometric area. In that case, the problem usually indicates how to handle the region.

Quick self-check

• Did you plug in both endpoints correctly?
• Did you keep parentheses around each substitution?
• Did you simplify the final expression exactly?

A good final answer is a clean exact value, such as an integer, fraction, or simplified expression.