Find the volume of the solid in the first octant bounded by the cylinder z = 16 - x² and the plane y = 5?

Solution:

The volume of the solid can be found using the triple integral.

V =\(\\\int_{0}^{2}\int_{0}^{2}\int_{0}^{4-x^{2}}dzdxdy\)

As the integrand is independent of y,

\(2\int_{0}^{2}\int_{0}^{4-x^{2}}dzdx\)

Now integrating with respect to z,

\(2\int_{0}^{2}z|_{z=0}^{z=4-x^{2}}dx\\=\int_{0}^{2}2(4-x^{2})dx\)

So the volume is

V = \(2(4x-\frac{1}{3}x^{3})|_{x=0}^{x=2}=2(8-\frac{8}{3})=\frac{32}{3}\)

Therefore, the volume of the solid is 32/3.

Find the volume of the solid in the first octant bounded by the cylinder z = 16 - x² and the plane y = 5?

Summary:

The volume of the solid in the first octant bounded by the cylinder z = 16 - x² and the plane y = 5 is 32/3.