Q. 6 - Calculus

Question

Q. Explain geometrically what the definition of the definite integral as a limit of Riemann sums represents. Include a labeled picture of a Riemann sum (for a particular n) that illustrates the roles of $n, ∆x, x_k, x_k^{*} \& f\left(x_k^{*}\right)$. What happens in the picture as $n\to \infty$?

Step-by-Step Solution

Answer

The final limit is ${\int }_0^1{x}^{2}dx=\underset{n\to \infty }{\lim }\sum _{k=1}^n{\left(\frac{k}{n}\right)}^2\left(\frac{1}{n}\right)$

Step 1. Given information

We have to express graphically the definition of the definite integral as a limit of Reimann sum

Step 2. Plotting graph

The graph  of $f\left(x\right)=x^2$ is

Step 3. Definite integral

The definite integral on the limits $x=a \to x=b$is

$$\begin{gathered} {\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k^{*}\right)∆x \\ \to ∆x=\frac{b-a}{n}, x_k=a+k∆x \& x_k^{*}=x_k \end{gathered}$$

Therefore the function $f\left(x\right)=x^2$on the interval $[0,1]$and any n rectangles.

$$\begin{gathered} ∆x=\frac{b-a}{n}=\frac{1}{n} \\ {x}_k=a+k∆x=\frac{k}{n} \\ {x}_k^{*}=x_k \end{gathered}$$

Taking the limits,

${\int }_0^1{x}^{2}dx=\underset{n\to \infty }{\lim }\sum _{k=1}^n{\left(\frac{k}{n}\right)}^2\left(\frac{1}{n}\right)$