Question

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.

\(\int_{{\rm{\pi /4}}}^{{\rm{5\pi /2}}} {{\rm{sin}}} {\rm{xdx}}\)

Short answer

\(\frac{{\sqrt {\rm{2}} }}{{\rm{2}}}\)

Step-by-step solution

Evaluating the definite integral.

\(\begin{aligned}{c}\int_{{\rm{\pi /4}}}^{{\rm{5\pi /2}}} {{\rm{sin}}} {\rm{xdx = ( - cosx)}}_{{\rm{\pi /4}}}^{{\rm{5\pi /2}}}\\{\rm{ = - }}\left( {{\rm{cos}}\left( {\frac{{{\rm{5\pi }}}}{{\rm{2}}}} \right){\rm{ - cos}}\left( {\frac{{\rm{\pi }}}{{\rm{4}}}} \right)} \right)\\{\rm{ = 0 + }}\frac{{\sqrt {\rm{2}} }}{{\rm{2}}}{\rm{ = }}\frac{{\sqrt {\rm{2}} }}{{\rm{2}}}\end{aligned}\)

Area illustrated by the figure below.

The definite integral can be interpreted as difference of area between \({\rm{5}}{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}\)and \({\rm{\pi }}\)and the area between\({\rm{\pi }}\) and \({{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{4}}}} \right.

\kern-\nulldelimiterspace} {\rm{4}}}\)

This can be illustrated by the figure below.