Find the area enclosed by the given curves.
-$y = \csc ^ { 2 } x , y = \cot ^ { 2 } x , x = \frac { \pi } { 4 }$, and $x = \frac { 3 \pi } { 4 }$
A) $\frac { \pi } { 4 }$
B) $\frac { 3 \pi } { 4 }$
C) $\pi$
D) $\frac { \pi } { 2 }$

Question

Quizplus · Accepted Answer

The area enclosed by the curves can be found by integrating the difference between the two functions from $x = \frac{\pi}{4}$ to $x = \frac{3\pi}{4}$. Since $y = \csc^2x$ and $y = \cot^2x$ are both positive and periodic in the given interval, and considering the trigonometric identity $\csc^2x = 1 + \cot^2x$, the area can be calculated as the integral of $(\csc^2x - \cot^2x)$ from $\frac{\pi}{4}$ to $\frac{3\pi}{4}$, which simplifies to the integral of $1$ over this interval. Thus, the area is $\frac{3\pi}{4} - \frac{\pi}{4} = \frac{\pi}{2}$.

Find the Area Enclosed by the Given Curves y=csc⁡2x,y=cot⁡2x,x=π4y = \csc ^ { 2 } x , y = \cot ^ { 2 } x , x = \frac { \pi } { 4 }y=csc2x,y=cot2x,x=4π​

Find the Area Enclosed by the Given Curves $y = \csc ^ { 2 } x , y = \cot ^ { 2 } x , x = \frac { \pi } { 4 }$