# 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.