Question

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

${\int }_0^{\frac{\mathrm{\pi }}{4}}{\int }_0^{secx}secxdydx$

Short answer

Value of the given integral is:$1$

Step-by-step solution

Step 1. Given Information 

An integral,${\int }_0^{\frac{\mathrm{\pi }}{4}}{\int }_0^{secx}secxdydx$

Step 2. Sketching the region of integration 

Region of integration is shaded in the graph below.

Step 3. Evaluating the given iterated integral  

Given integral,

$$\begin{gathered} {\int }_0^{\frac{\mathrm{\pi }}{4}}{\int }_0^{secx}secxdydx \\ ={\int }_0^{\frac{\mathrm{\pi }}{4}}{\overline{)\left(secx\right)\left(y\right)}}_0^{secx}dx \\ ={\int }_0^{\frac{\mathrm{\pi }}{4}}\left(secx\right)\left(secx-0\right)dx \\ ={\int }_0^{\frac{\mathrm{\pi }}{4}}\left(se{c}^{2}x\right)dx \\ ={\overline{)\left(\tan x\right)}}_0^{\frac{\mathrm{\pi }}{4}} \\ =\tan \left(\frac{\mathrm{\pi }}{4}\right)-\tan \left(0\right) \\ =1-0 \\ =1 \end{gathered}$$