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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=`

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Solution

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

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

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

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

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:

$$$f(x_{0})=f(a)=f(0)=1=1$$$

$$$2f(x_{1})=2f(\frac{1}{5})=2 \sqrt{\sin^{3}{\left (\frac{1}{5} \right )} + 1}=2.007826067912793$$$

$$$2f(x_{2})=2f(\frac{2}{5})=2 \sqrt{\sin^{3}{\left (\frac{2}{5} \right )} + 1}=2.058206972332648$$$

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

$$$2f(x_{4})=2f(\frac{4}{5})=2 \sqrt{\sin^{3}{\left (\frac{4}{5} \right )} + 1}=2.340214753424868$$$

$$$f(x_{5})=f(b)=f(1)=\sqrt{\sin^{3}{\left (1 \right )} + 1}=1.263258974474734$$$

Finally, just sum up above values and multiply by $$$\displaystyle\frac{\Delta{x}}{2}=\frac{1}{10}$$$: $$$\frac{1}{10}(1+2.007826067912793+2.058206972332648+2.17257446116512+2.340214753424868+1.263258974474734)=1.084208122931016$$$

Answer: $$$1.084208122931016$$$.


Related Note: Trapezoidal Rule