Q. 55 - Calculus

Question

Use the definition of the definite integral as a limit of Riemann sums to prove Theorem 4.11(b): For any function f that is integrable on $[a, b]$ and any real number c,

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

Step-by-Step Solution

Answer

The theorem 4.11(b)  is proved.

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

Step 1. Given Information

We are given a function f that is integrable.

Step 2. Proving the theorem

Proving the theorem,

$$\begin{gathered} {\int }_a^{b}cf\left(x\right)=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}cf\left(x_k^{*}\right)\Delta x \\ =c\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k^{*}\right)\Delta x \\ =c{\int }_a^{b}f\left(x\right) \end{gathered}$$

Hence Proved.