Q. 3 - Calculus

Question

Approximating with a Riemann sum: 

Use a right sum with $10$ rectangles to approximate the area under the graph of $f\left(x\right)=x^2$ on [0, 5].

Step-by-Step Solution

Answer

The area of the area under the graph $f\left(x\right)=x^2$ on the interval $[0,5]$ is $48.125$.

Step 1. Given Information.

The function:

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

Step 2. Graph the function on the graph.

Sketch the graph.

Step 3. Find the area.

The area is divided into $10$ rectangles. so, each rectangle is of length $∆x=0.5$.

Area, $$\begin{gathered} A=f(0.5)∆x+f\left(1\right)∆x+f(1.5)∆x+f\left(2\right)∆x+f(2.5)∆x+f\left(3\right)∆x+f(3.5)∆x+f\left(4\right)∆x+f(4.5)∆x+f\left(5\right)∆x \\ =0.25(0.5)+1(0.5)+2.25(0.5)+4(0.5)+6.25(0.5)+9(0.5)+12.25(0.5)+16(0.5)+20.25(0.5)+25(0.5) \\ =48.125 \end{gathered}$$