Q. 16 - Calculus

Question

Suppose $f$ is an integrable function $[a,b]$ and $k$ is a real number. Use pictures of Riemann sums to illustrate that the right sum for the function $kf\left(x\right) on [a,b] is k$  times the value of the right sum (with the same n) for $f$ on $[a,b]$. What happens as $n\to \infty$? What does this exercise say about the definite integrals ${\int }_a^{b}f\left(x\right)dx and {\int }_a^{b}kf\left(x\right)dx$?

Step-by-Step Solution

Answer

The definite integral ${\int }_a^{b}f\left(x\right)dx$ multiplied by $k$ - a real number will give the area under the function on the interval $[a,b]$ times $k.$

Step 1. Given information

Right sum approximation using rectangle of Riemann sums is given.

Step 2. Calculation

The right sum with $n$ rectangles of the function $f\left(x\right)$ on the interval $[a,b]$ can be evaluated as below

$∆x=\frac{b-a}{n}; x_k=a+w∆x: f\left(x_k\right)=a+w∆x$

$\sum _{w-1}^{n}f\left(x_w\right)∆x=$right sum approximation

If the above function is multiplied by $k$- a real number. Then the right sum of $kf\left(x\right)$ can be evaluated the same way.

$$\begin{gathered} ∆x=\frac{b-a}{n}; x_k=a+w∆x; kf\left(x_k\right)=k[a+w∆x] \\ \sum _{w-1}^{n}kf\left(x\right)∆x= right sum approximation \end{gathered}$$

Thus,

$\sum _{w-1}^{n}kf\left(x_w\right)=k\sum _{w-1}^{n}f\left(x_w\right)∆x$

Thus, the right sum of the function $kf\left(x\right) on [a,b]$ is equal to $k$ times the right sum of the function $f\left(x\right) on [a,b].$

As $n\to \infty$ The right sums approximation will converge to a constant value.

The definite integral ${\int }_a^{b}f\left(x\right)dx$ multiplied by $k-$a real number will give the area under the function on the interval $[a,b] times k.$

i.e. Area under the curve on $[a,b]*k=k{\int }_a^{b}f\left(x\right)dx$