Area Between Curves

3 questions to ask

  1. Where do we start? → upper bound

  2. Where do we end? → lower bound

  3. What function is on top? Does this change?

Example

Find the area of the region bounded by y = 2x + 5 and y = x3 between x = 0 and x = 2

Compare the functions within the interval [0, 2] to determine which is "on top”

at x = 0: y = 2(0) + 5 = 5 and y = 03 = 0

at x = 2: y = 2(2) + 5 = 9 and y = 23 = 8b

Since y = 2x + 5 is greater than y = x3 at both boundaries and they do not intersect between 0 and 2, the linear function is the upper function

The area A between two curves is found by integrating the difference (Upper - Lower)

Find the antiderivative

Evaluate the definite integral and apply the Fundamental Theorem of Calculus by plugging in the upper limit (2) and subtracting the lower limit (2)