Use a double integral to find the area enclosed by x = y − y^2 and x + y = 0.

Question

use double integral to solve this calculus based problem.

define the region and set the limits.

Answer 1

x = y - y^2

x+ y = 0

Rewriting the second equation as x in terms of y so that it has the same structure as the first equation and we can easily calculate the y limits, we get

x = y - y^2

x = -y

Thus, y - y^2 = -y

2y - y^2 = 0

y(2 - y) = 0

y = 0, 2

We are now ready to set up our double integral

∫₀²∫_-y^{y-y^2} dx dy =

∫₀² x|_-y^{y-y^2} dy =

∫₀² y - y^2 -(-y) dy =

∫₀² 2y - y^2 dy =

y^2 - y^3/3 |₀² =

2^2 - 2^3/3 - (0 - 0) =

4 - 8/3 =

4/3 or 1 1/3