Solutions to Quiz 4

1. Decide which of the following statements are true and which are false. No explanations required. (1pt each)

a) For a three dimensional object mass equals the product of average density and volume.

True

b) dV = dy dz dx. True

c) dA = r sin θ dr dθ. False, there is no sin θ in the formula.

d) ∫_{0}^{1} ∫_{1}^{y} tan(x + y) dy dx = 12π.

False, since after the first integration step there would still be y’s left in the formula and thus the answer wouldn’t be a number.

2. Find the mass of a circular disc of radius 3 centered at the origin whose density is given by the function δ(x, y) = x^{2} + y^{2}. (6pts)

∫_{0}^{3} ∫_{0}^{2π} r^{2} r dθ dr = ∫_{0}^{3} 2πr^{3} dθ dr = 81π/4.

3. Find the exact value of the following triple integral: (10pts)

∫_{0}^{1} ∫_{0}^{4} ∫_{√\overline{x}}^{2} sin(1 + y^{3})dy dx dz.

We will have to interchange the two innermost integrals:

∫_{0}^{1} ∫_{0}^{2} ∫_{0}^{y^{2}} sin(1 + y^{3})dx dy dz = ∫_{0}^{1} ∫_{0}^{2} y^{2} sin(1 + y^{3}) dy dz
= \frac{1}{3} ∫_{0}^{1} cos 1 − cos 9 dz = (cos 1 − cos 9)/3.