Question

If f(x) = ex − 1, 0 ≤ x ≤ 2, find the riemann sum with n = 4 correct to six decimal places, taking the sample points to be midpoints. m4 =

Answer 1

The Left Riemann sum uses the left endpoints of a subinterval:

$\int_{a}^{b}f(x)dx\approx \bigtriangleup x (f(x_0)+f(x_1)+2f(x_2)+...+f(x_{n-2}+f(x_{n-1})),where \bigtriangleup x=\frac{b-a}{n}$

We know that a=0, b=2, and n=4.

$\bigtriangleup x=\frac{b-a}{n}=\frac{2-0}{4}=\frac{1}{2}$

Divide the interval [0,2] into 4 subintervals of length $\bigtriangleup x=\frac{1}{2}$.

$a=[0,1/2],[1/2,1],[1,3/2],[3/2,2]$

$f(x_0)=f(a)=f(0)=0$

$f(x_1)=f(1/2)= 0.64872$

$f(x_2)=f(1)=1.71828$

$f(x_3)=f(3/2)=3.481689$

$\int_{0}^{2}f(x) dx= 1/2(0+0.64872+1.71828+3.481689)=2.9243$

Thus the Riemann sum of the given function $f(x)=e^x-1$ as 2.9243.