Q. 56 - Calculus

Question

Use the definition of the definite integral as a limit of Riemann sums to prove Theorem 4.12(a): For any function f and real number a,

${\int }_a^{a}f\left(x\right)dx=0$

Step-by-Step Solution

Answer

The theorem 4.12(a)  is proved. 

${\int }_a^{a}f\left(x\right)dx=0$

Step 1. Given Information

We are given a function f that is integrable. 

Step 2. Proving the theorem

Proving the theorem,

$$\begin{gathered} {\int }_a^{a}f\left(x\right)=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k^n\right)\Delta x \\ =\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k^{*}\right)0 \\ =0 \end{gathered}$$

Hence Proved.