Q. 8 - 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 right sum for $f\left(x\right)=\sqrt{x}$ on $\left[0,4\right] , n=8$

Step-by-Step Solution

Answer

The right sum is $3·17302$ .

Step 1. Given information .

Consider the given function $f\left(x\right)=\sqrt{x}$ .

Step 2. Formula used .

The formula for right sum is $S=\sum _{i=1}^n\delta x·f\left(x_i\right)$.

Step 3. Find the right sum .

$$\begin{gathered} f\left(x\right)=\sqrt{x} , \left[0,4\right] \\ \delta x=\frac{4-0}{n}=\frac{4}{n} \\ S=\underset{i=1}{\overset{8}{\sum \delta x·f\left(x_i\right)}} \\ =\delta x·f\left(0·5\right)+\delta x·f\left(1\right)+\delta x·f\left(1·5\right)+\delta x·f\left(2\right)+\delta x·f\left(2·5\right)+\delta x·f\left(3\right)+\delta x·f\left(3·5\right)+\delta x·f\left(4\right) \\ =0·5\times \sqrt{0·5}+0·5\times \sqrt{1}+0·5\times \sqrt{1·5}+0·5\times \sqrt{2}+0·5\times \sqrt{2·5}+0·5\times 2 \\ =3·17302 \end{gathered}$$