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 $[-\frac{\mathrm{\pi }}{2},2\mathrm{\pi }]$ ?

Short answer

three

Step-by-step solution

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 role="math" localid="1649148607017" $\left[-\frac{\mathrm{\pi }}{2},0\right]$

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

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

Therefore, the answer is three.