Exercise 256Calculus › Multivariable calculus › Multiple integralsHigh schoolAdded 3 Nov 2026

Double integral over a rectangle

#double integral#rectangle#iterated

Statement

Compute $\displaystyle I=\iint_D 2y\sin x\,dx\,dy$ where $D=[0,\pi]\times[0,1]$ .

Hint

The bounds are independent: integrate one variable at a time.

Show solutionHide solution

Iterated integral.
$I=\int_0^{\pi}\int_0^1 2y\sin x\,dy\,dx.$

Inner. $\int_0^1 2y\sin x\,dy=\sin x$ .

Outer. $I=\int_0^{\pi}\sin x\,dx=[-\cos x]_0^{\pi}=2$ .

Conclusion. $I=2$ .