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Wize University Calculus 1 Textbook > Integration Techniques

The Midpoint Rule

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The Midpoint Rule

We can begin to get more creative with our Reimann Sums to get better approximations for area under curves. Instead of using a left or right hand approximation, we can place our rectangles so the function touches the midpoint of the rectangle.


Midpoint Rule

∫abf(x)dx ≈∑i=1nf(xi‾)Δx=Δx(f(x1‾)+f(x2‾)+f(x3‾)+...+f(xn−1‾)+f(xn‾))\boxed{\displaystyle \int_a^bf\left(x\right)dx\ \approx\sum_{i=1}^nf\left(\overline{x_i}\right)\Delta x=\Delta x\left(f\left(\overline{x_1}\right)+f\left(\overline{x_2}\right)+f\left(\overline{x_3}\right)+...+f\left(\overline{x_{n-1}}\right)+f\left(\overline{x_n}\right)\right)}

where Δx=b−an,  xi=a+iΔx,  xi‾=xi−1+xi2\displaystyle \Delta x=\frac{b-a}{n}, \ \ x_i=a+i\Delta x , \ \ \overline{x_i}=\frac{x_{i-1}+x_i}{2} and nn is the number of subintervals.

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Example: The Midpoint Rule

Use the midpoint approximation for

∫04ex2dx\displaystyle \int_0^4e^{x^2}dx

using 4 sub intervals.


∫abf(x)dx ≈∑i=1nf(xi‾)Δx=Δx(f(x1‾)+f(x2‾)+f(x3‾)+...+f(xn−1‾)+f(xn‾))\displaystyle \int_a^bf\left(x\right)dx\ \approx\sum_{i=1}^nf\left(\overline{x_i}\right)\Delta x=\Delta x\left(f\left(\overline{x_1}\right)+f\left(\overline{x_2}\right)+f\left(\overline{x_3}\right)+...+f\left(\overline{x_{n-1}}\right)+f\left(\overline{x_n}\right)\right)


Δx=b−an=4−04=1\displaystyle \Delta x=\frac{b-a}{n}=\frac{4-0}{4}=1

∫04ex2dx≈1(f(.5)+f(1.5)+f(2.5)+f(3.5)) =(e.52+e1.52+e2.52+e3.52)\displaystyle \int_0^4e^{x^2}dx\approx1\left(f\left(.5\right)+f\left(1.5\right)+f\left(2.5\right)+f\left(3.5\right)\right)\ =\left(e^{.5^2}+e^{1.5^2}+e^{2.5^2}+e^{3.5^2}\right)

=(e.25+e2.25+e6.25+e12.25)=\left(e^{.25}+e^{2.25}+e^{6.25}+e^{12.25}\right)