Question

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

Short answer

The volume of the given solid can be:

\(V = \frac{{17}}{{60}}\).

Step-by-step solution

Find the limits

Consider the following plane:\(x - 2y + z = 1\)

Rewrite the given plane as:\(z = 1 - x + 2y\)

The region D can be obtained by the following graph.

From the above figure, the region D can be written as:\(D = \{ (x,y)|0 \le x \le 1,1 - x \le y \le 1 - {x^2}\)

The volume under the plane \(x - 2y + z = 1\) and above D is:

Find the integral

Hence, The volume of the given solid can be evaluated as: \(V = \frac{{17}}{{60}}\).