Question

use the given partition and sample points to approximate the definite integral of $f(x)=x^{2}+5x + 8$ on the indicated interval. $x_0=-3<-2<-1<0<1=x_4,x_i^*=x_{i - 1}$

Explanation:

Step1: Recall Riemann - sum formula

$R_n=\sum_{i = 1}^{n}f(x_{i}^*)\Delta x_i$

Step2: Calculate $\Delta x_i$

$\Delta x_1=-2-(-3)=1,\Delta x_2=-1 - (-2)=1,\Delta x_3=0-(-1)=1,\Delta x_4=1 - 0=1$

Step3: Find sample - points and function values

$x_1^*=-3,f(x_1^*)=(-3)^2+5(-3)+8=9 - 15 + 8 = 2$;
$x_2^*=-2,f(x_2^*)=(-2)^2+5(-2)+8=4 - 10 + 8 = 2$;
$x_3^*=-1,f(x_3^*)=(-1)^2+5(-1)+8=1 - 5 + 8 = 4$;
$x_4^*=0,f(x_4^*)=0^2+5(0)+8=8$.

Step4: Calculate Riemann - sum

$R_4=2\times1+2\times1+4\times1+8\times1=16$

Answer:

$16$