# Questions in Section Calculus III: page 2

Re-write the equations in cylindrical coordinates:

1. (x+y)(x-y)+2y^2=z-2.

2. xsqrt(x^2+y^2)=z.

Find the Jacobian of the transformation when x=uv, y=vw, z=uw.

Evaluate the following multiple integral: iiint_E xdV where E is the volume bounded by z=4-x^2-y^2 and z=x^2+3y^2 and in the first octant (x>=0, y>=0, z>=0).

Evaluate the following multiple integral: iint_D x^2 y dA where D is the region between y=sqrt(x), y=1/x, and x=3.

Evaluate the following integral in Spherical coordinates:

iiint_B sqrt(x^2+y^2+z^2)dV, where B is the region with x^2+y^2+z^2<=2z.

Solve the following integral using Polar coordinates:

iint_D (x+y)dA, where D is bounded by the x-axis, y-axis and circles x^2+y^2=4 and x^2+y^2=16 and is in the first quadrant.

Solve the following integral using Cylindrical coordinates:

iiint_E zsqrt(x^2+y^2)dV over the cylinder x^2+y^2<=4 for 1<=z<=5.

The depth of water in a swimming pool fits the equation f(x,y)=2sin(x/20-7)-3cos((x-3)/5)+8 when 0<=x<=20, and the sides of the pool fit the equations y(x)=10-((x-10)^2)/10 and y(x)=((x-10)^2)/20-5.

Evaluate the volume under the curve f(x,y)=2xcos(y^3), over the region D, bounded by the line y=x and the y-axis, when 0<=y<=2.

Estimate the following integrals with double/triple Riemann sums using the midpoint rule.

1. iint_R(5x-2y^2+3xy)dA where R={(x,y)|1<=x<=2,3<=y<=5} and m=2,n=4.
2. iiint_E (4x+4y-4z)dV where E={(x,y,z)|0<=x<=3,0<=y<=3,0<=z<=3} and m=n=l=2.