Q. 81 - Calculus

Question

Suppose f is an integrable function on [a, b] and $x_k = a+k\left(\frac{b-a}{n}\right).$

(a) Use the definition of the definite integral as a limit of Riemann sums to show that

$\underset{n\to \infty }{\lim }\frac{\sum _{k=1}^{n}f\left(x_k\right)}{n} = \frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx.$

(b) Why is it algebraically sensible that the left-hand side of the equation is a calculation of average value?

(c) Why is it graphically sensible that the right-hand side of the equation is a calculation of average value?

Step-by-Step Solution

Answer

(a) Hence, Proved.

(b) It is because left-hand side is nothing but the average value of $f\left(x_k\right)$.

(c) It is because right-hand side is nothing but the area of the graph f(x) in the interval [a, b] divided by the length of the interval. 

Step 1. Given Information.

f is an integrable function on [a, b] and $x_k = a+k\left(\frac{b-a}{n}\right).$

Step 2. part (a) Proof.

$$\begin{gathered} To prove that: \underset{n\to \infty }{\lim }\frac{\sum _{k=1}^{n}f\left(x_k\right)}{n} = \frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx. \\ \underset{n\to \infty }{\lim }\frac{\sum _{k=1}^{n}f\left(x_k\right)}{n} \\ = \underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k\right)\left(\frac{1}{n}\right) \\ = \frac{1}{b-a}\left(\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k\right)\left(\frac{b-a}{n}\right)\right) \\ = \frac{1}{b-a}\left(\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k\right)\left(∆x\right)\right) \\ = \frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx. \\ Hence, proved. \end{gathered}$$

Step 3. part (b) Average value.

The left hand side is nothing but the sum of $f\left(x_1\right), f\left(x_2\right), f\left(x_3\right), .... f\left(x_k\right)$ divided by the total number of terms. This is the same as like the average value of the set of numbers.

Step 4. part (c) Area under the curve.

The right-hand side is the area under the curve f(x) in the interval of [a, b] divided by the length of the interval i.e., (b-a) which is the average value of the function in the given interval.