Double Integrals
The double integral of 1 over a region is used to find this.
What is area?
You can order a triple integral this many ways.
What is 6?
This is the conversion of f(x,y) = x + 4x^2 + 4y^2 + y into polar coordinates.
r(cost+sint) + 4r^2
This is the reversal of the order of integration of the double integral with bounds (0->lnx)dy and (1->2)dx
(e^y->2)dx, (0->ln2)dy
This type of double integral region is bounded by two continuous f(x) and two constants.
What is Type I?
This is how you would change a double integral into an equivalent triple integral.
Take the integrand and have z go from 0 to it.
These are the three conversions of (x, y, z) into spherical coordinates.
What are psin(phi)cos(theta), psin(phi)sin(theta), pcos(phi)
This is an integral to describe the volume of the region between z=1-x^2, y=1-x, z=0, y=0 and x=0.
various (show graph)
The double integral over a circle of radius 1 with the integrand f(x,y)=x is this.
What is 0?
This is the equation for the z-coordinate of the center of mass of an object in 3d.
What is the integral of z times density/mass?
The equation of a semicircle in x-y coordinates is y=this
What is sqrt(a2-x2)?
This is the volume of the ball p <= 3 between the cones phi=pi/6 and phi=pi/3.
9pi(sqrt(3)-1)
This is the equation for the x-coordinate of the CoM of an object.
What is the integral of x times the density/the mass?
This is the integral that describes the volume of the shape with vertices (0, 0, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1), (0, 1, 0) and (0, 0, 1).
These are the bounds of an integral to describe a cylinder of radius 2 and height 3, where x > 0.
-pi/2 > theta > pi/2
0 > r > 2
0 > z > 3
This is the x coord of the center of mass of the section of a disk of radius 2 in the third quadrant with density d(x, y) = xy
What is -16/15?
This is the volume below the plane 3x+2y-z=0 and above the region bounded by y=x^2 and x=y.
What is 23/60?
This is the equation to describe the mass of a region shaped like the unit ball with density at any point equal to that point's distance from the origin
What is integral (0-pi) (0-2pi) (0-1) (p^3)sintheta dp dtheta dphi?
If the integrand of a triple integral in cartesian coordinates is x(x^2+y^2-xz^2)+z, then the integrand in spherical coordinates is...
p^3sin^3(phi)cos(theta)-p^4sin^2(phi)cos^3(phi)cos(theta) + pcos(phi)
The integral of x+y+z bounded by the graphs of z=x^2-1, z=1-x^2, y=2 and y=0.
What is 16/3?