Chapter 6.2, Problem 10E

To determine

To evaluate: The given integral

Answer to Problem 10E

The value of the given integral is $\frac{3\pi }{2}$

Explanation of Solution

Given Information:

The integral ${\displaystyle \underset{-4}{\overset{-1}{\int }}\frac{\pi }{2}d\theta }$

Formula used:

The integration of $d\theta$is given by,

  ${{\displaystyle {\int }_a^{b}d\theta =\left[\theta \right]}}_a^b$

Calculation The given expression represents a definite integral where the differential variable $d\theta$is to be integrated with the limits of integration being [-4, -1].

The integration of $d\theta$is given by,

  ${{\displaystyle {\int }_a^{b}d\theta =\left[\theta \right]}}_a^b$

Considering the given integral,

  $\begin{array}{c}{\displaystyle \underset{-4}{\overset{-1}{\int }}\frac{\pi }{2}d\theta }=\frac{\pi }{2}{\displaystyle \underset{-4}{\overset{-1}{\int }}d\theta }\\ =\frac{\pi }{2}{\left[\theta \right]}_{-4}^{-1}\\ =\frac{\pi }{2}\times \left(-1-\left(-4\right)\right)\\ =\frac{3\pi }{2}\end{array}$

Conclusion: The value of the integral ${\displaystyle \underset{-4}{\overset{-1}{\int }}\frac{\pi }{2}d\theta }$ is $\frac{3\pi }{2}$