MAC 2313  Homework 3Suppose one corner of a 10’x10’ rectangular pool
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Updated on: 04/10/2016 04:19 PM Due on: 05/10/2016
MAC 2313  Homework 3
Answer each question with justification for the main steps. Solutions and/or formulas without
explanation will be worth zero points. You are expected to turn in a finished product. You are
encouraged to discuss these problems between yourselves or with me during office hours.
1. (4 points) Suppose one corner of a 10’x10’ rectangular pool is at the origin and the bottom surface of
the pool is given by
.
(a) Express the exact volume of the pool as an iterated integral. Do not try to evaluate it.
(b) Approximate the volume using Riemann sums with 2,5, and 10 subdivisions for both axes and
midpoint samples.
(c) The above integral can’t be explicitly computed. If you needed the exact volume to within 2 decimal
places, you might continue computing Riemann sums with more subdivisions. Explain how you
might decide when you had taken enough to achieve the required accuracy. Justify your answer.
2. (2 points) Assume r > 0 is a constant. Find the average minimum distance between a point in the solid
sphere {(x,y,z) ∈R3 : x2 + y2 + z2 ≤ r2} and the zaxis.
3. (2 points) Find the point (a,b) for which the volume between the plane 2x + 6y + 2z = 12 and the z = 0
plane is maximized over the rectangle R = [0,a] × [0,b]. It may help to sketch the region for a few choices
of (a,b).
4. (2 points) Exercise 22 from section 15.6.
Answer each question with justification for the main steps. Solutions and/or formulas without
explanation will be worth zero points. You are expected to turn in a finished product. You are
encouraged to discuss these problems between yourselves or with me during office hours.
1. (4 points) Suppose one corner of a 10’x10’ rectangular pool is at the origin and the bottom surface of
the pool is given by
.
(a) Express the exact volume of the pool as an iterated integral. Do not try to evaluate it.
(b) Approximate the volume using Riemann sums with 2,5, and 10 subdivisions for both axes and
midpoint samples.
(c) The above integral can’t be explicitly computed. If you needed the exact volume to within 2 decimal
places, you might continue computing Riemann sums with more subdivisions. Explain how you
might decide when you had taken enough to achieve the required accuracy. Justify your answer.
2. (2 points) Assume r > 0 is a constant. Find the average minimum distance between a point in the solid
sphere {(x,y,z) ∈R3 : x2 + y2 + z2 ≤ r2} and the zaxis.
3. (2 points) Find the point (a,b) for which the volume between the plane 2x + 6y + 2z = 12 and the z = 0
plane is maximized over the rectangle R = [0,a] × [0,b]. It may help to sketch the region for a few choices
of (a,b).
4. (2 points) Exercise 22 from section 15.6.

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Solution: MAC 2313  Homework 3Suppose one corner of a 10’x10’ rectangular pool