Polar, cylindrical, and spherical coordinates

1 Polar, cylindrical, and spherical coordinates

Using polar coordinates, compute
\[ \int _{-4}^0 \int _{- \sqrt {16 - x^2}} ^0 y \, dy dx \]
This is a quarter of the disk of radius 4
Using polar coordinates, compute
\[ \int _{0}^{3/ \sqrt {2}} \int _{ y } ^{\sqrt {9 - y^2 }} x^2y \, dx dy \]
This is a sector with angle \(\pi /4\) of the circle of radius 3
Using polar coordinates, compute
\[ \iint _B x \, dx dy \]
where \(B\subset \mathbb {R}^2\) is the region in the first quadrant inside the circle \(x^2 + (y-3)^2 = 9\)
This is a quarter of the unit disk
Using polar coordinates, show that
\[ \int _{- \infty } ^{\infty } e^{- \frac {x^2}{2} } \, dx = \sqrt {2 \pi } \]
The square of the integral on the left is \(\iint _{\mathbb {R}^2} e^{- \frac {x^2 + y^2 }{2}} \, dA \).
Let \(D \subset \mathbb {R}^2\) be the interior of the circle \((x-1)^2 + (y-1)^2 = 2\). Using polar coordinates, find
\[ \iint _D \frac {x}{x^2 + y^2 } dA \]
The notes have a section on how to describe a region like this in polar coordinates.
Using cylindrical coordinates, compute
\[ \int _0 ^3 \int _ {- \sqrt {9 - y^2} } ^{\sqrt {9 - y^2}} \int _ y ^{4} x ^2 \, dz dxdy \]
Deal separately with the \(x,y\) variables and the \(z\) variable. For cylindrical coordinates, most of the time you don’t need to do anything to the \(z\) variable
Using cylindrical coordinates, compute
\[ \int _0 ^ 1 \int _ {- 1 } ^{1} \int _z ^{ \sqrt {1 - y ^2 }} y^2 \, dx dy dz \]
This is the region inside the cylinder \(x^2 + y^2 = 1\), above the \(xy\)-plane, and below the plane \(z = x \). Verify this!!!
Using spherical coordinates, compute
\[ \int _{-2} ^2 \int _ { - \sqrt {4 - x^2 } } ^{\sqrt {4 - x^2}} \int _ {0} ^{\sqrt {4 - x^2 - y^2 }} y^2 \, dz dydx \]
This is a portion of the ball of radius 2
Using spherical coordinates, compute
\[ \iiint _ E zy^2 \, dV \]
where \(E\) is the region above the cone \(z = - \sqrt {x^2 + y^2 }\) and inside the sphere \(x^2 + y^2 + z^2 = 25 \)
The equation of the cone is \(\phi = 3 \pi / 4\) and the equation of the sphere is \(\rho = 5\)
Using spherical coordinates, compute
\[ \iiint _ E \frac { z x^2}{x^2 + y^2} \, dV \]
where \(E\) is the region below the plane \(z = 3\) and above the cone \(z = \sqrt {x^2 + y^2} / \sqrt {3}\)
The equation of the plane is \( \rho = 3 / \cos \phi \)
Using spherical coordinates, compute
\[ \iiint _ E z \, dV \]
where \(E\) is the region inside the sphere \(x^2 + y^2 + (z-3)^2 = 9 \)
The notes have a section on how to describe a region like this in polar coordinates.