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`.
  1. «
  2. 1
  3. 2
  4. 3
  5. 4
  6. 5
  7. »