Question

Integral represents the volume of a solid. Describe the solid

\(\int_{\rm{0}}^{{\rm{\pi /4}}} {\rm{2}} {\rm{\pi (\pi - x)(cos x - sin x)dx}}\).

Short answer

The volume of integral in the given equation \({\rm{y = cosx,y = sinx,x = 0,}}\) about line\({\rm{x = \pi }}\).

Step-by-step solution

Volume of the solid axis \({\rm{x}}\)and \({\rm{y}}\).

When a curve is rotated above the y-axis, it is said to be "rotated above the y-axis."

The solid obtained has a volume of

\({\rm{V = }}\int_{\rm{a}}^{\rm{b}} {\rm{2}} {\rm{\pi xydx}}\)

When a curve is rotated above the x-axis.

The solid obtained has a volume of

\({\rm{V = }}\int_{\rm{a}}^{\rm{b}} {\rm{2}} {\rm{\pi xydy}}\).

Solid of given Integral.

Given integral,

\({\rm{V = 2\pi }}\int_{\rm{0}}^{{\rm{\pi /4}}} {{\rm{(\pi - x)(cosx - sinx)}}} {\rm{dy}}\)

One possible interpretation of the integral would be the region bounded by

\({\rm{y = cosx,y = sinx,x = 0,}}\)about line\({\rm{x = \pi }}\).

Another way \({\rm{y = cosx - sinx,y = 0}}\).

Therefore, the solid volume boundary is \({\rm{y = cosx,y = sinx,x = 0,}}\) about line\({\rm{x = \pi }}\).