Q. 14 - Calculus

Question

Calculate each definite integral in using

Part (a): The definition of the definite integral as a limit of Riemann sums.

Part (b): The definite integral formulas from Theorem 4.13.

Part (c): the Fundamental Theorem of Calculus. Then show that your three answers are the same.

${\int }_0^4{\left(3x+2\right)}^2 dx$

Step-by-Step Solution

Answer

Part (a): 

Part (b): 

Part (c): 

Part (a) Step 1. Given information.

Consider the given question,

${\int }_0^4{\left(3x+2\right)}^2 dx$

Part (a) Step 2. Using limit of Riemann sums.

The right sum defined for n rectangles on $\left[a,b\right]$ is $\sum _{k=1}^{n}f\left(x_k\right)△x$.

Where, $$\begin{gathered} △x=\frac{b-a}{n}, \\ {x}_k=a+k△x \end{gathered}$$.

The interval is $\left[4,0\right]$. Now $△x$,

$$\begin{gathered} =\frac{4-0}{n} \\ =\frac{4}{n} \end{gathered}$$

Then $x_k$,

$$\begin{gathered} =1+k\left(\frac{4}{n}\right) \\ =1+\frac{4k}{n} \end{gathered}$$

Part (a) Step 3. Write the right sum.

Consider the right sum,

$$\begin{gathered} =\sum _{k=0}^n{\left[ 3\left(1+\frac{4k}{n}\right)+2\right]}^2 \\ =\sum _{k=0}^n{\left[ {\displaystyle \frac{5n+12k}{n}}\right]}^2 \\ =\frac{25n^2}{n^2}\sum _{k=0}^n{\left(\frac{12k}{n}\right)}^2 \end{gathered}$$