Q. 3 - Calculus

Question

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

Will the absolute area be positive or negative, and why? Will the signed area will be positive or negative, and why? 

Step-by-Step Solution

Answer

three

positive

negative

Step1. Given Information

The function is,

$f\left(x\right)=\sin x$

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

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

  Consider, the following graph

$\text{ From the above figure it is clear that }3\text{ intervals are required. }$

Step2. Determining Area

The objective is to determine if the absolute area will be negative or positive. 

The absolute area will be positive as the area covered will be positive.

Step3. Determining signed area

$\text{ The objective is to determine if the signed area will be positive or negative. }$

$\text{ The signed area will be negative because the signed area on }[0,\pi ]\text{ and }[\pi ,2\pi ]\text{ are equal and }$

  opposite.

$\text{ Therefore, the answers are three, positive and negative }$