Question

evaluate double integral cos ((π/2)x^2)) dxdy, along y=x, x=1, and along the x axis.

Answer 1

\[\int\limits_0^1 \int\limits _0^x \cos(\frac{\pi}{2} x^{2}) dy dx\]
\[= \int\limits_0 ^1 x \cos(\frac{\pi}{2} x^{2}) dx\]
\[u = \frac{\pi}{2} x^{2} \rightarrow du = \pi x dx\]
\[= \frac{1}{\pi} \int\limits_0^1 \cos(u) du\]
\[= \frac{1}{\pi}(\sin \frac{\pi}{2} -\sin 0)\]
\[=\frac{1}{\pi}\]