Chapter 5.7, Problem 1E

(a)

To determine

The integral by trapezoidal rule with four intervals.

Answer to Problem 1E

The value of integral by trapezoidal rule is $10.75$ .

Explanation of Solution

Given:

The integral is ${\displaystyle \underset{1}{\overset{3}{\int }}\left(x^2+1\right)dx}$ .

Calculation:

Integrate the function by trapezoidal rule as follows:

  $\underset{a}{\overset{b}{{\displaystyle \int }}}f(x)dx\approx \frac{\text{Δ}x}{2}\left(f(x_0)+2f(x_1)+2f(x_2)+\mathrm{...}+2f(x_{n-1})+f(x_n)\right)$

We have, $a=1,b=3,n=4$

Therefore,

  $\begin{array}{c}\Delta x=\frac{3-1}{4}\\ =\frac{1}{2}\end{array}$

End points of integral are calculated as follows:

  $\begin{array}{l}f\left(x_0\right)=f(a)=f\left(1\right)=2=2\\ 2f\left(x_1\right)=2f\left(\frac{3}{2}\right)=\frac{13}{2}=6.5\\ 2f\left(x_2\right)=2f\left(2\right)=10=10\\ 2f\left(x_3\right)=2f\left(\frac{5}{2}\right)=\frac{29}{2}=14.5\\ f(x_4)=f(b)=f(3)=10=10\end{array}$

Substitute all the values in trapezoidal integral equation as follows:

  $\begin{array}{c}\underset{a}{\overset{b}{{\displaystyle \int }}}f(x)dx\approx \frac{1}{4}(2+6.5+10+14.5+10)\\ =10.75\end{array}$

Thus, the value of integral by trapezoidal rule is $10.75$ .

(b)

To determine

The integral by simpson’s rule with four intervals.

Answer to Problem 1E

The value of integral by simpson’s rule is $10.67$ .

Explanation of Solution

Given:

The integral is ${\displaystyle \underset{1}{\overset{3}{\int }}\left(x^2+1\right)dx}$ .

Calculation:

Integrate the function by simpson’s rule as follows:

  $\underset{a}{\overset{b}{{\displaystyle \int }}}f(x)dx\approx \frac{\text{Δ}x}{3}\left(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+\mathrm{...}+2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right)$ We We have, $a=1,b=3,n=4$

Therefore,

  $\begin{array}{c}\Delta x=\frac{3-1}{4}\\ =\frac{1}{2}\end{array}$

End points of integral are calculated as follows:

  $\begin{array}{l}f\left(x_0\right)=f(a)=f\left(1\right)=2=2\\ 4f\left(x_1\right)=4f\left(\frac{3}{2}\right)=13=13\\ 2f\left(x_2\right)=2f\left(2\right)=10=10\\ 4f\left(x_3\right)=4f\left(\frac{5}{2}\right)=29=29\\ f(x_4)=f(b)=f(3)=10=10\end{array}$

Substitute all the values in simpson’s integral equation as follows:

  $\begin{array}{c}\underset{a}{\overset{b}{{\displaystyle \int }}}f(x)dx\approx \frac{1}{6}(2+13+10+29+10)\\ =10.67\end{array}$

Thus, the value of integral by simpson’s rule is $10.67$ .