This is the given integral, which is currently in rectangular coordinates, how would I rewrite this in spherical and cylindrical coordinate systems?
2026-03-31 13:40:32.1774964432
How would I take a given triple integral in rectangular coordinates and rewrite it in cylindrical and spherical coordinates?
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Hint:
$\sqrt{x^2 + y^2} \leq z \leq 1$, $\, 0 \leq y \leq \sqrt{1 - x^2}, 0 \leq x \leq 1$
So it is a cone with vertex at the origin and $z = r$ (consider inverted cone). The upper bound of $z$ is $1$ and you see $x^2 + y^2 \leq 1$. You should also consider the lower bound of $x$ and $y$ which means you are only integrating over the region in first octant.
So the bounds in cylindrical coordinates, for example, will be straightforward
$r \leq z \leq 1$, $\, 0 \leq r \leq 1, 0 \leq \theta \leq \frac{\pi}{2}$
Can you now try and do the same for spherical coordinates? Just think what happens as you change the polar angle and what does it do to the limits of integration? Till certain polar angle, $\rho$ will be defined by the top of your cone and beyond that by the ray of the cone. $z = r$ will play an important role in defining the limits of polar angle and splitting your integral. Please sketch so it becomes clear.