Chapter 15.4, Problem 48E

To determine

To find: the value of the given integral without using limits or the Fundamental Theorem of calculus.

Answer to Problem 48E

  ${\displaystyle \underset{2}{\overset{5}{\int }}\left(3x-6\right)}=\frac{27}{2}$ .

Explanation of Solution

Given:

Thegiven function is ${\displaystyle \underset{2}{\overset{5}{\int }}\left(3x-6\right)}$ .

Calculation:

As thegiven function is ${\displaystyle \underset{2}{\overset{5}{\int }}\left(3x-6\right)}$ .

According to the question, this is not solved by using limits or the Fundamental Theorem of calculus.

So, the equation is solved by graphical method.

Therefore, the graph of ${\displaystyle \underset{2}{\overset{5}{\int }}\left(3x-6\right)}$ is:

  

From the graph, it is concluded that the area (red region) is the value of the integral.

Therefore, ${\displaystyle \underset{2}{\overset{5}{\int }}\left(3x-6\right)}=\text{ area of the red region}=\frac{1}{2}\left(5-2\right)9=\frac{27}{2}$

Hence,the derivative of the given functionis ${\displaystyle \underset{2}{\overset{5}{\int }}\left(3x-6\right)}=\frac{27}{2}$ .