Question

Use a triple integral to find the volume of the given solid. The solid enclosed by the cylinder y=x^2 and the planes z=0 and y+z=1.

Answer 1

The volume of the solid enclosed by the cylinder y = x^2 and the planes z = 0 and y + z = 1 is 7 / 30.

Step-by-step explanation:

To find the volume of the solid enclosed by the cylinder y = x^2 and the planes z = 0 and y + z = 1, we can set up a triple integral over the region that describes the solid.

The region is defined by 0 leq z leq 1, 0 leq y leq x^2, and y + z leq 1.

The triple integral for the volume (V) is given by:

V = int_{0}^{1} int_{0}^{x^2} int_{0}^{1 - y} dz , dy , dx

Now, evaluate the integral:

V = int_{0}^{1} int_{0}^{x^2} (1 - y), dy, dx

V = int_{0}^{1} [y - y^2 / 2]_{0}^{x^2}, dx

V = int_{0}^{1} [x^2 - x^4 / 2], dx

V = [{x^3} / {3} - x^5 / 10]_{0}^{1}

V = 1 / 3 - 1 / 10 - (0 - 0)

V = 7 / 30

So, the volume of the solid is 7 / 30.