Question

Give a geometric argument to prove Theorem 4.13(a): For any real numbers a, b, and c,

${\int }_a^{b}cdx=c(b-a)$

(Hint: Use a rectangle.)

Short answer

The theorem 4.13(a) is proved.

${\int }_a^{b}cdx=c(b-a)$

Step-by-step solution

Step 1. Given Information 

We are given a theorem,

${\int }_a^{b}cdx=c(b-a)$

Step 2. Proving the theorem 

Take, $f\left(x\right)=c$on $[a,b]$. Then the integral ${\int }_a^{b}f\left(x\right)dx$, represents the area from a to b under the constant curve $f\left(x\right)=c$. That is, the integral ${\int }_a^{b}f\left(x\right)dx$, represents the area of the rectangle of length $(b-a)$ and breadth c.

Hence Proved.

${\int }_a^{b}f\left(x\right)dx=c(b-a)$.