eMathHelp
Contribute Ask Question Register   Login

Math Notes

Home / Calculators / Calculus II Calculators

Trapezoidal Rule Calculator

Calculator will approximate integral using trapezoidal rule.

Enter a function: `f=`

Enter lower limit: `a=`

Enter upper limit: `b=`

Enter number of rectangles: `n=`

Write all suggestions in comments below.


Show steps



Solution

Your input: approximate integral $$$\int_{0}^{1}\sqrt{\sin^{3}{\left (x \right )} + 1}\ dx$$$ using $$$n=5$$$ rectangles.

Trapezoidal rule states that $$$\int_{a}^{b}f(x)dx\approx\frac{\Delta{x}}{2}\left(f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)\right)$$$, where $$$\Delta{x}=\frac{b-a}{n}$$$.

We have that $$$a=0$$$, $$$b=1$$$, $$$n=5$$$.

Therefore, $$$\Delta{x}=\frac{1-0}{5}=\frac{1}{5}$$$.

Divide interval $$$\left[0,1\right]$$$ into $$$n=5$$$ subintervals of length $$$\Delta{x}=\frac{1}{5}$$$ with the following endpoints: $$$a=0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1=b$$$.

Now, we just evaluate function at those endpoints:

$$$f\left(x_{0}\right)=f(a)=f\left(0\right)=1=1$$$

$$$2f\left(x_{1}\right)=2f\left(\frac{1}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{1}{5} \right )} + 1}=2.00782606791279$$$

$$$2f\left(x_{2}\right)=2f\left(\frac{2}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{2}{5} \right )} + 1}=2.05820697233265$$$

$$$2f\left(x_{3}\right)=2f\left(\frac{3}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{3}{5} \right )} + 1}=2.17257446116512$$$

$$$2f\left(x_{4}\right)=2f\left(\frac{4}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{4}{5} \right )} + 1}=2.34021475342487$$$

$$$f\left(x_{5}\right)=f(b)=f\left(1\right)=\sqrt{\sin^{3}{\left (1 \right )} + 1}=1.26325897447473$$$

Finally, just sum up above values and multiply by $$$\frac{\Delta{x}}{2}=\frac{1}{10}$$$: $$$\frac{1}{10}(1+2.00782606791279+2.05820697233265+2.17257446116512+2.34021475342487+1.26325897447473)=1.08420812293102$$$

Answer: $$$1.08420812293102$$$.


Related Note: Trapezoidal Rule