So I am given the following parametric equations.
$$ y=bsin(\theta) $$ and $$ x=acos(\theta) $$
When I do the following I get a negative area. $$ \int_0^{2\pi}b\sin\theta\frac{d}{d\theta}\left(a\cos\theta\right)d\theta$$
I looked on slader and they were doing the following integral:
$$ \int_{\frac{\pi}{2}}^0b\sin\theta\left(\frac{d}{d\theta}\left(a\cos\theta\right)\right)d\theta $$
Can someone please explain to me why they did this? Also is taking the absolute value appropriate in cases like this?
The formula for the are under a positive graph and the x-axis is defined to be $$\int _a ^b f(x)dx$$ Where $a<b$ and $f(x)\ge 0$
Here $dx$ is positive to make the integral positive.
We have to be sure that in our parametric integral for the area, $$\int b \sin \theta d(a\cos \theta)$$ we have our $d(a\cos (\theta)\ge 0$ to get a positive answer.
That is why they swapped the limits of integration.