Q. 21 - 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.) 

${\int }_1^{4}3\left(2^x\right)dx$

Step-by-Step Solution

Answer

The required value is $\frac{42}{\ln \left(2\right)}$.

Step 1. Given Information

We are given the definite integral ${\int }_1^{4}3\left(2^x\right)dx$ 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:   

$$\begin{gathered} {\int }_1^{4}3\left(2^x\right)dx \\ =3{\int }_1^4\left(2^x\right)dx \\ =3{\left[\frac{1}{\ln \left(2\right)}{2}^x\right]}_1^4 \\ =\frac{3}{\ln 2}\left(2^4-2^1\right) \\ =\frac{3}{\ln 2}\left(16-2\right) \\ =\frac{3}{\ln 2}\left(14\right) \\ =\frac{42}{\ln 2} \end{gathered}$$

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 $\frac{42}{\ln \left(2\right)}$ square units.