Area Between Curves Practice Problems with Answers

These area between curves practice problems start with simple regions and move on to crossing curves and integrating with respect to y. Try each one first. The answers come first so you can check quickly, then the full solutions.

The formula:

Find where the curves meet, decide which one is on top, and integrate the difference. For a longer explanation with pictures, read the area between curves guide.

Problems

Two curves, one region

1. y = x and y = x^2
2. y = 4 - x^2 and y = 0
3. y = 2x and y = x^2
4. y = x^2 - 4 and y = 2 - x
5. y = \sqrt{x} and y = x^2

Given limits

6. y = \cos x and y = \sin x from x = 0 to x = \frac{\pi}{4}
7. y = e^x and y = 1 from x = 0 to x = 1
8. y = x and y = \frac{1}{x} from x = 1 to x = 2

Harder setups

9. The region bounded by x = y^2 and x = y + 2
10. The total area between y = x^3 and y = x on [-1, 1]

Answers

Step-by-step solutions

1. Set x = x^2: x = 0 or x = 1. At x = \frac{1}{2}, the line gives 0.5 and the parabola 0.25, so the line is on top:

2. The parabola meets the x-axis at x = \pm 2:

3. Set 2x = x^2: x = 0 or x = 2. At x = 1, the line is higher:

4. Set x^2 - 4 = 2 - x: x^2 + x - 6 = 0, so x = -3 or x = 2. At x = 0, the line gives 2 and the parabola -4, so the line is on top:

5. The curves meet at x = 0 and x = 1, and \sqrt{x} is on top in between:

6. On \left[0, \frac{\pi}{4}\right], \cos x \geq \sin x:

7. On [0, 1], e^x \geq 1:

8. For x > 1, x > \frac{1}{x}:

9. Both curves are written as x in terms of y, so integrate in y. Set y^2 = y + 2: y = -1 or y = 2. At y = 0, the line gives x = 2 and the parabola x = 0, so the line is on the right:

10. The curves meet at x = -1, 0, 1. On [0, 1], x \geq x^3. On [-1, 0], x^3 \geq x. By symmetry the two pieces are equal:

Integrating x - x^3 over [-1, 1] in one go gives 0, which is wrong.

Common mistakes on these problems

• Bottom minus top. A negative answer is the giveaway. Test a point to find the top curve.
• Not splitting at a crossing. In problem 10, one integral cancels the two regions and gives 0.
• Using the wrong limits. The limits come from where the curves meet, not from the x-intercepts of one curve.
• Mixing x and y. In problem 9, the integrand and the limits must both be in y.

Keep practicing

For unlimited regions with instant feedback, try the area between curves lesson and the applications practice set. Next, spin these regions around an axis in the volumes of revolution problems.