Question

Evaluate the double integral:\(\iint\limits_D {xdA,D = \{ (x,y)/0 \leqslant x \leqslant \Pi ,0 \leqslant y \leqslant sinx\} }\)

Short answer

value of integral is \(\Pi \)

Step-by-step solution

Step 1:- Considering x as constant, & integrating with respect to y.

\(\begin{array}{l}I = \int\limits_0^\Pi {\int\limits_0^{sin(x)} {xdydx} } \\ = \int\limits_0^\Pi {\int\limits_0^{sin(x)} {dydx.} } \end{array}\)

Step 2:- Integrate on y.

\(\begin{array}{l}I = \int\limits_0^\Pi {x(y)} \left| {_0^{sinx}dx} \right.\\ = \int\limits_0^\Pi {xsin(x)dx.} \end{array}\)

Using integration by parts

\(I = - xcos(x)\left| {_0^\Pi + \int\limits_0^\Pi {cos(x)dx} } \right.\)

\(\begin{array}{l} = - \Pi ( - 1) - 0 + sin(x)\left| {_0^\Pi } \right.\\ = \Pi + (0 - 0)\\ = \Pi \end{array}\)

Hence, value of integral is \(\Pi \)