Question

\(\int\limits_0^{\frac{\pi }{2}} {\int\limits_0^{cos x} {f(x,y)dy} dx} \)

Short answer

We should know how to do graphical representation.

Step-by-step solution

Graphical representation:

The inner limits of integration indicate that \(0 \le y \le \cos (x)\). This gives two of the boundary lines: \(y = 0\) and \(y = \cos (x)\).

The outer limits of integration indicate that \(0 \le x \le \frac{\pi }{2}\).

Graph these four boundaries and select the interior as shown in the given below.

Graph:

Changing order of integration

A horizontal line through this region enters on the left at \(x = 0\) and reaches the right boundary at \(y = \cos (x)\) or \(x = {\cos ^{ - 1}}(y)\). The extreme \(y\)values are \(0\) and \(1\).

Therefore, the given integral can be written: \(\int\limits_0^1 {\int\limits_0^{{{\cos }^{ - 1}}(y)} {f(x,y)dx} dy} \).