Questions in Section Calculus III

True or False?

A sphere with center `P(x_0, y_0, z_0)` and radius `r` consists of all points `(x, y, z)` that satisfy the inequality `(x-x_0)^2+(y-y_0)^2+(z - z_0)^{2} <= r^2`.

True or False?

If a point belongs to both the xy-plane and the xz-plane, then the point lies on the x-axis.

True or False?

The graph of `x^2+y^2=1` in 3-space is a circle of radius 1 centered at the origin.

True or False?

The line through the points `(-3, 8,-4)` and `(1, 0, 8)` intersects the yz-plane at `(0,2,5)`.

Find the parametric equation of the line through `(0,3)` that is parallel to the line `x=-5+t, y=1-2t`.

Find the point on the line segment joining `P_1=(1, 4,-3)` and `P_2=(1, 5,-1)` that is `2/3` of the way from `P_1` to `P_2`.

We need to construct a water channel with aluminium material. We have a piece of rectangular aluminium plate with width 24 cm. As shown in Figure, the channel will be formed with a shape of isosceles trapezoid (with an open topside).

Evaluate the line integral `int_C[(4y+5)dx+2xzdy+(yz-x)dz]` along the following paths `C`:

  1. `C` is defined parametrically by `x=2t^2`, `y=6t`, `z=t^3` from `t=0` to `t=1`.
  2. `C` consists of straight lines from `(0,0,0)` to `(0,1,0)` then to `(0,1,1)` and then to `(6,1,1)`.
  3. `C` is given by the straight line joining `(0,0,0)` to `(1,2,6)`.

For the double integral `I=int_0^6 int_y^6 xy sqrt(1+x^4)dxdy`:

  1. sketch the region of integration;
  2. reverse the order of integration and then evaluate the integral analytically.

Re-write the equations in cylindrical coordinates:

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

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

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