Q. 10 - 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 trapezoid sum for $f\left(x\right)=9-x^2$ on $\left[0,3\right] , n=6$

Step-by-Step Solution

Answer

The trapezoid sum is $17·8$ .

Step 1. Given information .

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

Step 2. Formula used .

${\int }_a^{b}f\left(x\right) dx =\frac{\delta x}{2}\left[f\left(x_0\right)+f\left(x_n\right)+2\left[f\left(x_i\right)+f\left(x_2\right)+.............+f\left(x_{n-1}\right)\right]\right]$

Step 3. Find the trapezoid sum .

Trapezoid sum for $f\left(x\right)=9-x^2$ on $\left[0,3\right] , n=6$ .

$\delta x=\frac{3-0}{6}=0·5$

$x$	$0$	$0·5$	$1$	$1·5$	$2$	$2·5$	$3$
$f\left(x\right)$	$9$	$8·75$	$8$	$6·75$	$5$	$2·75$	$0$

$$\begin{gathered} =2\left[f\left(x_1\right)+f\left(x_2\right)+.............+f\left(x_{n-1}\right)\right] \\ =2\left[8·75+8+6·75+5+2·75\right] \\ =62·50 \\ {\int }_0^{3}f\left(x\right) dx =\frac{1}{4}\left[9+0+62·5\right] \\ =17·8 \end{gathered}$$