Q. 11 - Calculus

Question

Riemann sums: Calculate the following Riemann sum

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

given value of n .

The upper sum for $f\left(x\right)=4x-x^2$ on $\left[0,3\right]$ , $n=3$

Step-by-Step Solution

Answer

The sum is 10 .

Step 1. Given information .

Consider the given function $f\left(x\right)=4x-x^2$ on $\left[0,3\right]$ .

Step 2. Formula used .

The Riemann sum formula for upper sum is $U\left(P,f\right)=\sum _{r=0}^3{M}_r\delta x_r$ .

Step 3. Find the value .

Divide open interval $\left[0,3\right]$ in three sub intervals $\left(0,1\right) , \left(1,2\right) ,\left(2,3\right)$ where M is 1 , 2 , 3 substitute the value of M in the given function to find $M_{1 }, M_2, M_3$.

$M_1=4x-x^2=4·1-1=3$

$M_2=4 , M_3=3$

Further simplify .

$$\begin{gathered} \delta x_1=1-0=1 \\ \delta x_2=2-1=1 \\ \delta x_3=3-2=1 \end{gathered}$$

Substitute the values of M and $\delta x$in the given formula .

$$\begin{gathered} U\left(P,f\right)=\sum _{r=0}^3{M}_r\delta x_r \\ =M_1\delta x_1+M_2\delta x_2+M_3\delta x_3 \\ =3\times 1+4\times 1+3\times 1 \\ =10 \end{gathered}$$

$M_1=4x-x^2=4\times 1-1=3$