Can someone help me with these problems and explain each step. Any help is appreciated.
Find the area of the regions enclosed by the lines and curves.
x = (tany)^2 and x = -(tany)^2, -Π/4 ≤y ≤ Π/4
Can someone help me with these problems and explain each step. Any help is appreciated.
Find the area of the regions enclosed by the lines and curves.
x = (tany)^2 and x = -(tany)^2, -Π/4 ≤y ≤ Π/4
Hello,
the enclosed area is calculated by the difference of functions:
$\displaystyle x_1=\tan^2(y)$ and $\displaystyle x_2=-\tan^2(y)$
$\displaystyle x_1-x_2=2\tan^2(y)$
The enclosed area is:
$\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}{2\tan^2(y) dy} = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}{2\cdot \frac{\sin^2(y)}{\cos^2(y)} dy}$$\displaystyle =2\cdot \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}{ \frac{1-\cos^2(y)}{\cos^2(y)} dy}=2\cdot \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}{ \left(\frac{1}{\cos^2(y)}-1 \right) dy}=$
$\displaystyle \left[2 \cdot \tan(y) - 2y \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}}=\left(2-\frac{\pi}{2}\right)-\left( -2-\frac{-\pi}{2} \right)=4-\pi$
I've attached a diagram of this area.
EB