Q. 8 - Calculus

Question

Draw pictures illustrating the fact that if $a\le c\le b$ then

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

Step-by-Step Solution

Answer

The graphs are as follows

Step 1. Given information

An identity is given as ${\int }_a^{c}f\left(x\right)dx+{\int }_c^{b}f\left(x\right)dx={\int }_a^{b}f\left(x\right)dx$

Step 2. Graphs

Draw the of individual functions and then compare them.

The first graph of $f\left(x\right)=x^2 \& from 0\to 0.5$

Now the second graph of $f\left(x\right)=x^2 \& from 0.5\to 1$

Last graph of $f\left(x\right)=x^2 \& from 0\to 1$

From the above graphs we can conclude that

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