Q. 24 - Calculus

Question

Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals in Exercises 19–64. Use a graph to check your answer. (Hint: The integrands that involve absolute values will have to be considered piecewise.) 

${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\sin \left(3x\right)dx$

Step-by-Step Solution

Answer

The required value is $0$.

Step 1. Given Information

We are given the definite integral ${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\sin \left(3x\right)dx$ and we need to use the Fundamental Theorem of Calculus to find the exact value of the integral.    

Step 2. Finding the integral

The required value is: 

$$\begin{gathered} {\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\sin \left(3x\right)dx \\ =\frac{1}{3}{\left[-\cos \left(3x\right)\right]}_{-\mathrm{\pi }}^{\mathrm{\pi }} \\ =\frac{1}{3}\left(-\cos \left(3\mathrm{\pi }\right)+\cos \left(-3\mathrm{\pi }\right)\right) \\ =\frac{1}{3}\left(-\cos \left(3\mathrm{\pi }\right)+\cos \left(3\mathrm{\pi }\right)\right) \\ =0 \end{gathered}$$

Step 3. Rechecking solution

The required graph is:   

After plotting the graph we can see that the area under the graph is exactly same as the area obtained from the definite integral. The area under the graph is $0$ square units.