Midpoint Rule Calculator for a Function

An online calculator for approximating the definite integral using the midpoint (mid-ordinate) rule, with steps shown.

Related calculator: Midpoint Rule Calculator for a Table

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Your Input

Approximate the integral $$$\int\limits_{1}^{3} \sqrt{\sin^{4}{\left(x \right)} + 7}\, dx$$$ with $$$n = 4$$$ using the midpoint rule.

Solution

The midpoint rule (also known as the midpoint approximation) uses the midpoints of a subinterval:

$$$\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \Delta x \left(f{\left(\frac{x_{0} + x_{1}}{2} \right)} + f{\left(\frac{x_{1} + x_{2}}{2} \right)} + f{\left(\frac{x_{2} + x_{3}}{2} \right)}+\dots+f{\left(\frac{x_{n-2} + x_{n-1}}{2} \right)} + f{\left(\frac{x_{n-1} + x_{n}}{2} \right)}\right)$$$

where $$$\Delta x = \frac{b - a}{n}$$$.

We have that $$$f{\left(x \right)} = \sqrt{\sin^{4}{\left(x \right)} + 7}$$$, $$$a = 1$$$, $$$b = 3$$$, and $$$n = 4$$$.

Therefore, $$$\Delta x = \frac{3 - 1}{4} = \frac{1}{2}$$$.

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

Now, just evaluate the function at the midpoints of the subintervals.

$$$f{\left(\frac{x_{0} + x_{1}}{2} \right)} = f{\left(\frac{1 + \frac{3}{2}}{2} \right)} = f{\left(\frac{5}{4} \right)} = \sqrt{\sin^{4}{\left(\frac{5}{4} \right)} + 7}\approx 2.794821922941848$$$

$$$f{\left(\frac{x_{1} + x_{2}}{2} \right)} = f{\left(\frac{\frac{3}{2} + 2}{2} \right)} = f{\left(\frac{7}{4} \right)} = \sqrt{\sin^{4}{\left(\frac{7}{4} \right)} + 7}\approx 2.817350905627184$$$

$$$f{\left(\frac{x_{2} + x_{3}}{2} \right)} = f{\left(\frac{2 + \frac{5}{2}}{2} \right)} = f{\left(\frac{9}{4} \right)} = \sqrt{\sin^{4}{\left(\frac{9}{4} \right)} + 7}\approx 2.714130913751178$$$

$$$f{\left(\frac{x_{3} + x_{4}}{2} \right)} = f{\left(\frac{\frac{5}{2} + 3}{2} \right)} = f{\left(\frac{11}{4} \right)} = \sqrt{\sin^{4}{\left(\frac{11}{4} \right)} + 7}\approx 2.649758163512828$$$

Finally, just sum up the above values and multiply by $$$\Delta x = \frac{1}{2}$$$: $$$\frac{1}{2} \left(2.794821922941848 + 2.817350905627184 + 2.714130913751178 + 2.649758163512828\right) = 5.488030952916519.$$$

Answer

$$$\int\limits_{1}^{3} \sqrt{\sin^{4}{\left(x \right)} + 7}\, dx\approx 5.488030952916519$$$A