Question

Which of the following values would be obtained using 5 midpoint rectangles of equal width to estimate the integral from 0 to 1 of x squared, dx?

Answer 1

Partition the interval [0,1] as

$\left[0,\dfrac15\right]\cup\left[\dfrac15,\dfrac25\right]\cup\left[\dfrac25,\dfrac35\right]\cup\left[\dfrac35,\dfrac45\right]\cup\left[\dfrac45,5\right]$

The midpoints of the intervals are, respectively $\dfrac1{10},\dfrac3{10},\dfrac12,\dfrac7{10},\dfrac9{10}$ - these are your sample points.

The integral is approximated by

$\displaystyle\int_0^1x^2\,\mathrm dx\approx\sum_{n=1}^5f(x_n)\Delta x$

where $\Delta x$ is the difference between the partition endpoints, i.e. $\Delta x=\dfrac{1-0}5=\dfrac15$, and $x_n$ is the midpoint of the $n$th partition. You have

$\displaystyle\sum_{n=1}^5f(x_n)\Delta x=\frac15\left(\left(\frac1{10}\right)^2+\left(\frac3{10}\right)^2+\left(\frac12\right)^2+\left(\frac7{10}\right)^2+\left(\frac9{10}\right)^2\right)=\dfrac{33}{100}$

For comparison, the actual value of the integral is $\dfrac13$, so the approximation is valid to two decimal places.