Applications of Integrals — Practice

Question 1

Find the area bounded by y = x^2 + 2, the x-axis, x = 0, and x = 3.

Accepted Answer

A = integral 0 to 3 of (x^2 + 2) dx = [x^3/3 + 2x] from 0 to 3 = 9 + 6 = 15 square units.

Question 2

Find the area bounded by y^2 = x and y = x.

Accepted Answer

Intersections at x = 0 and x = 1, with sqrt(x) >= x on [0,1]. A = integral 0 to 1 of (sqrt(x) - x) dx = [2x^(3/2)/3 - x^2/2] = 2/3 - 1/2 = 1/6 square units.

Question 3

Find the area of the ellipse x^2/9 + y^2/4 = 1 by integration.

Accepted Answer

a = 3, b = 2, so area = pi a b = 6 pi square units.

Question 4

Find the area bounded by y = sin x and the x-axis from x = 0 to 2 pi.

Accepted Answer

sin x is positive on [0, pi] and negative on [pi, 2pi]. A = integral 0 to pi of sin x dx + |integral pi to 2pi of sin x dx| = 2 + 2 = 4 square units.

Question 5

Find the area between y = 4 - x^2 and y = x^2 - 4.

Accepted Answer

Intersections at x = +/- 2 with (4 - x^2) above. A = integral -2 to 2 of (8 - 2x^2) dx = [8x - 2x^3/3] from -2 to 2 = 64/3 square units.

Practice — Applications of Integrals

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