Q. 4 - Calculus

Question

Without using absolute values, how many definite integrals would we need in order to calculate the area between the graphs of f(x) = sin x and g(x) = $\frac{1}{2}$on $[-\frac{\mathrm{\pi }}{2},2\mathrm{\pi }]$ ? 

Step-by-Step Solution

Answer

three

Step1. Given Information

$\text{ The functions are, }$

$f\left(x\right)=\sin x;g\left(x\right)=\frac{1}{2}$

$\text{ The objective is to determine the number of definite integrals would require in order to calculate }$

$\text{ the absolute area between }f\left(x\right)=\sin x\text{ and }g\left(x\right)=\frac{1}{2}\text{ on }\left[-\frac{\pi }{2},2\pi \right]\text{. }$

Step2. Consider, the following graph

Step3. Answer

From the above figure it is clear that 3 intervals are required to calculate  the absolute area between $f\left(x\right)=\sin x$ and $g\left(x\right)=\frac{1}{2}$ on  $\left[-\frac{\mathrm{\pi }}{2},2\mathrm{\pi }\right]$

1st inteval from $\left[-\frac{\mathrm{\pi }}{2},0\right]$

2nd interval $\left[0, \mathrm{\pi }\right]$

3rd interval $\left[\mathrm{\pi },2\mathrm{\pi }\right]$

Therefore, the answer is three.