The Midpoint Rule is more accurate than the Trapezium Rule. It works by finding the mid-points of rectangles drawn to the curve. The Midpoint Rule is:
![{\displaystyle \int _{a}^{b}f\left(x\right)\,dx\approx =h\left[f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots +f\left(x_{n}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8892220f2181ca22226fdf533fd77b93b9ac525)
Where: 
n is the number of strips.
and ![{\displaystyle x_{i}={\frac {1}{2}}\left[\left(a+\left\{i-1\right\}h\right)+\left(a+ih\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5259213f3fabd4478a2a51ec53b732043773d266)
Use the Midpoint Rule to evaluate 
using 4 strips.
Firstly, we work out h.
Now we begin to set up the Midpoint Rule.
![{\displaystyle \int _{1}^{5}x^{2}+2x\ dx\approx 1\left[f\left(x_{1}\right)+f\left(x_{2}\right)+f\left(x_{3}\right)+f\left(x_{4}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83fb7e84f5051cfc2d0c1bae3b1bc18b5d10b927)
![{\displaystyle x_{1}={\frac {1}{2}}\left[\left(1+\left\{1-1\right\}1\right)+\left(1+1\times 1\right)\right]={\frac {1}{2}}\left[\left(1\right)+\left(2\right)\right]=1.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0febdfaefb63016865bd0a89abb96415598d544e)
![{\displaystyle x_{2}={\frac {1}{2}}\left[\left(1+\left\{2-1\right\}1\right)+\left(1+2\times 1\right)\right]={\frac {1}{2}}\left[\left(2\right)+\left(3\right)\right]=2.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/477f6d74bcae9c72a10e7505ca1d0c631791cedf)
![{\displaystyle x_{3}={\frac {1}{2}}\left[\left(1+\left\{3-1\right\}1\right)+\left(1+3\times 1\right)\right]={\frac {1}{2}}\left[\left(3\right)+\left(4\right)\right]=3.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8144919e14e5f263f246fdf0c01d6f67c71a3beb)
![{\displaystyle x_{4}={\frac {1}{2}}\left[\left(1+\left\{4-1\right\}1\right)+\left(1+4\times 1\right)\right]={\frac {1}{2}}\left[\left(4\right)+\left(5\right)\right]=4.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb550382401add9e7b845ab9a10887b16bac217)
![{\displaystyle \int _{1}^{5}x^{2}+2x\ dx\approx 1\left[f\left(1.5\right)+f\left(2.5\right)+f\left(3.5\right)+f\left(4.5\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b838f12b606af5a409618516836ab4201f43f5f6)
Now we need to solve f(n)
![{\displaystyle \int _{1}^{5}x^{2}+2x\ dx\approx 1\left[5.25+11.3+19.3+29.3\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5c61d53cda7b10461408d351ca335bbd45a46d)
As you can see the resultant from the midpoint rule is closer to the true value 
than the trapezium rule, but worse than Simpson's Rule.
