I want to calculate the volume of a sphere with radius $r$ using calculus. I can split this sphere into lots of great circles of that sphere and then add all the areas to find the volume. Let's write
$$ V = \int_{0}^{\pi} \pi r^2 d \theta = \pi^2r^2 $$
But we all know the volume should be $\frac{4}{3} \pi r^3$. So, what's wrong with this?
You actually want to calculate $\int_{-r}^r \pi(r^2-x^2) dx$, where by the Pythagorean theorem $r^2-a^2$ is the squared radius of the circle at $x=a$. In polar coordinates, another option is $\int_0^{2\pi}d\phi\int_0^\pi \sin\theta d\theta\int_0^r \rho^2 d\rho$.