Question

Find the volume of the given solid. Under the plane \(z = 1 + {x^2}{y^2}\)and above the region bounded by \(x = {y^2}\)and\(x = 4\)

Short answer

The volume of the given solid can be:

\(V = \frac{{2336}}{{27}}\).

Step-by-step solution

Find the limits

Consider the surface\(z = 1 + {x^2}{y^2}\)and above the region enclosed by\(x = {y^2}\)and\(x = 4\)

The integrand should be equally easy whether integrated first on x or first on y. the region of integration is type 1 and also type 2.

Therefore integrate first on x in order to maintain whole number exponents. The region is graphed below along with the horizontal line to help set up the integral.

The region D can be obtained by the following graph.

Note that the left end of the horizontal line is on the curve \(x = {y^2}\)and the right end is on the line\(x = 4\).

The horizontal line can sweep from \(y = - 2\)to\(y = 2\).

Therefore the integral can be set up as:

Find the integral

Hence, The volume of the given solid can be evaluated as: \(V = \frac{{2336}}{{27}}\).