Question

Use a limit of Riemann sums to calculate the exact area under the graph of $f\left(x\right)=x^2$on $[0,5]$.

Short answer

The area of the area under the graph $f\left(x\right)=x^2$on the interval $[0,5]$is $41.6$.

Step-by-step solution

Step 1. Given Information.

The function:

$f\left(x\right)=x^2$ on the interval $[0,5]$

Step 2. Riemann integral formula.

The area can be estimated by,

$A={\int }_a^{b}f\left(x\right).dx$

Step 3. Substitute the values and find the area.

Substitute the known values in the above formula, we get,

$$\begin{gathered} A={\int }_0^5{x}^{2}dx \\ ={{\left[\frac{x^3}{3}\right]}^5}_0=\frac{1}{3}(5^3-0^3) \\ =\frac{125}{3} \\ \approx 41.6 \end{gathered}$$