Questions in Section Calculus III: page 5

Find the total charge `Q` of the wire if the line density is distributed along the wire according to the formula: `q(x,y,z)=x^2+y^2-z^2`. The vector equation of the wire is `r=4cos hati+(3t-5)hatj+4sinthatk`, where `0<=t<=3`. (All quantities here are dimensionless.)

Find the mass of the plate `D` bounded by the given lines if the surface density of the plate `p(x,y)` `[g/(cm^2)]` is the known function: `p(x,y)=(2x+5y)/(x^2+y^2)`, and the plate `D` is bounded by the following lines: `x^2+y^2=9`; `x^2+y^2=16`; `x=0`; `y=0`; `(x>0,y>0)`.

Using double integration find the area of the domain bounded by the following lines: `y=3/2sqrt(x)`, `y=3/(2x)`, when `x` varies between the point of line intersection (find it) and `x=9`. Make a sketch and show the domain.

Evaluate the work done by the vector field `I=int_Gamma[(e^xy+cosxsiny)dx+(e^x+sinxcosy)dy]` between the points `A(pi/3,pi/2)` and `B((5pi)/6,(3pi)/4)`.

Make an independent decision about the most convenient path of integration `Gamma`. Present your result both in the exact and approximate forms (up to two decimal places in the latter case).

A tent has volume `V=6m^3`. The tent has the shape shown in the figure with ends but no floor. It is desired to make the tent with the least possible material.

  1. Derive a formula for the volume, `V`, in terms of `w`, `L` and `theta`. Then rewrite this formula with `L` as the subject.
  2. Derive a formula for the area of material, `A`, in terms of `w`, `L` and `theta` and then substitute for `L`. Given that `V` is a constant, this gives `A` as a function of the 2 variables, `w` and `theta`.
  3. Find the two partial derivatives, `(del A)/(del w)` and `(del A)/(del theta)`.
  4. Find the stationary point (subject to the constraints that `w>0` and `0<theta<pi/2`). Substitute the values `w=w_0`, `theta=theta_0`, say, as given by the stationary point for the volume of the tent specified above, into the formula for `L` in part a) to find the corresponding value `L=L_0`.
  5. Finally, use the second derivatives test (or otherwise) to show that the stationary point of part d) is a local minimum.

Re-write the equations in spherical coordinates:

  1. `(x^2+y^2+z^2)/(x-y)=9z`.

  2. `x^2-y^2=z`.

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