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 $kf\left(x\right)on[a,b]isk$ 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${\int }_a^{b}f\left(x\right)dxand{\int }_a^{b}kf\left(x\right)dx$?

Short 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-by-step solution

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.

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

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 $[a,b]timesk.$

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