Q. 58 - Calculus

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.)

Step-by-Step Solution

Answer

The theorem 4.13(a)  is proved. 

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

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.

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