Question

Calculate the iterated integral.

\(\int_0^{\pi /2} {\int_0^y {\int_0^x {cos} } } (x + y + z)dzdxdy\)

Short answer

The value of the integral is \(\frac{{ - 1}}{3}\)

Step-by-step solution

Concept

The integral which is inside is the one the whole integral needs to be integrated with respect to and consequently the next integral.

Use substitution to evaluate the integral

The integral is \(\int_0^{\pi /2} {\int_0^y {\int_0^x {cos} } } (x + y + z)dzdxdy\)

\(\begin{aligned}\int_0^{\frac{\pi }{2}} {\int_0^y {\int_0^x {\cos } } } (x + y + z)dzdxdy = \int_0^{\frac{\pi }{2}} {\int_0^y {\sin } } (2x + y) - \sin (x + y)dxdy = \\\int_0^{\frac{\pi }{2}} {\frac{{2\cos 2y - \cos 3y + \cos y - 2\cos y}}{2}} dy = \frac{{ - 1}}{3}\end{aligned}\)

Hence thevalue of the iterated integral is \(\frac{{ - 1}}{3}\)_Z