Multiple Integrals

Double Integrals in General Regions and Rectangles

100

What are the two ways of solving a double integral over a general region?

Vertical and horizontal strips

200

∬3dA; -2 < x< 2, 1<y<6

60

300

Set up but don't evaluate the integral for the region under the surface z=xy and above the triangle with vertices (1,1), (4,1), (1,2)

∬xy dy dx

y from y=1 to y=-1/3x+4/3

400

Evaluate the double integral enclosed by the surface z=xsec²(y) and the planes z=0, x=0, x=2, y=0, and y=π/4

2

500

Set up, but don't evaluate, a double integral for the region enclosed by the parabolic cylinders y=1-x², y=x²-1, and the planes x+y+z=2 and 2x+2y-z+10=0.

∬(10+2x+2y)-(2-x-y)dydx

x from -1 to 1 and y from 1- x² to x²-1 

Double and Triple Integrals in Polar and Cylindrical Regions

100

What do you have to remember to add when converting an integral from standard to polar form?

To multiply by r

200

Convert x²+y2+2x+3y to polar

r²+2rsinΘ+3sinΘ

300

Set up but do not evaluate the triple integral for the volume of the solid bounded bounded by the planes: z=x+5, z=0 and the cylinder y²+x²=9

400

Set up but don't evaluate The ∬e^(-(x²)-(y²))dA bounded by the semicircle x=√(4-y²) and the y axis.

∬e^{-r^2} r drdΘ 

0<r<2 -pi/4<Θ<pi/4

500

Set up do not evaluate the ∭x²dV bounded by the the cylinder x²+y²=1 and below the plane z=1 above the cone z² = 4x²+4y²

∭r³cos²Θ dz dr dΘ

2r<z<1 0<r<1 0<Θ<2pi

Applications of double integrals

100

What is the formula for center of mass for x and y coordinates (x,y) notation

(My/m, Mx/m)

200

What is the surface area formula

A(S) = ∬D√1+(ꝺf/ꝺx)2+ (ꝺf/ꝺy)2dA OR magnitude of r_u x r_v dA

300

Find the mass of the disk given by x²+ y²>= 16 in the xy-plane with the density function ρ(x, y) = x²

64π

400

Find the area of the surface of the part of the plane 3x + 2y + z = 6 that lies in the first octant

3 √(14)

500

Find the center of mass ∬ydA where the region is bounded by y=e^x, y=0, x=0, and x=1

((e²+1)/2(e²-1),  4(e³-1)/9(e²-1)

Triple Integrals in Spherical Coordinates

100

What is z equal to in spherical coordinates?

z = ⲣcosɸ

200

What is x equal to in spherical coordinates?

x = ⲣsinɸcosΘ

300

Convert (2, π/4, π/3) from spherical to cartesian coordinates.

(√(6)/2,(√(6)/2,1)

400

where the surface is a ball with a center at the origin and a radius of 5.

312,500π/7

500

Find the center of mass of ∬ydA where the region is bounded by y=e^x,y=0, x=0, and x=1.

( (e²+1)/2(e²-1), 4(e³-1)/9(e²-1) )

Change in Multiple Variables

100

What is the purpose of a Jacobian?

To correct for the spatial disruption caused by the change of variables.

200

Why do you parmaetrize functions in terms of u and v?

To make the function 'nicer' and easier to integrate

300

What is the formula needed for solving a question involving a Jacobian?

∬sf(g(u,v), h(u,v))|ꝺ(x,y)/ꝺ(u,v)|dudv

400

Find the Jacobian of the transformation x=uv/v, y=v/w, z=u/w

0

500

Evaluate ∬(x-2y)/(3x-y)dA where R is the parallelogram enclosed by the lines x-2y=0, x-2y=4, 3x-y=1, and 3x-y=8

8/5(ln8)