Math 225: Calculus III    Quiz 7    3/25&27/97

<<Calculus`Math225`

Calculate the integral of f[x_,y_] = Sin[Pi(x^2+y^2)] over the unit disk,
x^2 + y^2 <=1.

Solution

Use polar coordinates: x^2 + y^2 == r^2. the region is desribed by
    0 <=   r   <= 1
    0 <= theta <= 2 Pi
Recall, we must add a factor of r into the integral for the area element, dA == r dr d(theta). In fact, this extra r makes the integral easy by allowing the substitution u == r^2, du == 2r dr.

Integrate[ Sin[Pi r^2] r, {theta,0,2Pi}, {r,0,1}]

Find the average height of points in the bottom corner of a unit cube. The corner is defined to be the solid region below the plane x + y + z == 1 in the first octant. Its volume is 1/6. The height function is, of course, f[x_,y_,z_] = z.

Solution

vol = Integrate[1, {x,0,1}, {y,0,1-x}, {z,0,1-x-y}]

int = Integrate[z, {x,0,1}, {y,0,1-x}, {z,0,1-x-y}]

zbar = int / vol

t = Graphics3D[Polygon[{{1,0,0},{0,1,0},{0,0,1}}]];
s = Graphics3D[Polygon[{{0,0,0},{1,0,0},{0,0,1}}]];

Show[t,s]