Q. 9 - Calculus

Question

Riemann sums: Calculate each of the following Riemann sum

approximations for the definite integral of f on [a, b], using the

given value of n.

The midpoint sum for $f\left(x\right)=9-x^2$ on $\left[0,3\right] , n=3$ .

Step-by-Step Solution

Answer

The midpoint sum is $18·25$ .

Step 1. Given information .

Consider the given function $f\left(x\right)=9-x^2$ .

Step 2. Find the midpoint sum .

Midpoint sum for $f\left(x\right)=9-x^2$ on $\left[0,3\right] , n=3$ .

$$\begin{gathered} a=0 , b=3 , n=3 \\ \delta x=\frac{3-0}{3}=1 \end{gathered}$$

Sub interval	Midpoints	$f\left(x_i\right)$	$\delta x_i·f\left(x_i\right)$
$0-1$	$0·5$	$9-{\left(0·5\right)}^2=8·75$	$8·75$
$1-2$	$1·5$	$9-{\left(1·5\right)}^2=6·75$	$6·75$
$2-3$	$2·5$	$9-{\left(2·5\right)}^2=2·75$	$2·75$

$$\begin{gathered} {\int }_0^{3}f\left(x\right) dx=6·75+8·75+2·75 \\ =18.25 \end{gathered}$$