MATH 138 Assignment 5
1. The frustrum of a regular triangular pyramid is depicted below. The top equilateral triangle has side length a, the base equilateral triangle has side length b, and the height of the frustrum is h.
(a) Set up an integral to calculate the volume of the frustrum.
(b) Evaluate your integral to show that the volume of the frustrum is 3h (a2 + ab + b2 ).
12
MC questions
2. The region between two curves y = f(x) and y = g(x) (where f(x) ≥ g(x) ≥ 0) on [2,5] is rotated about the line x = 9 to form a solid S. The volume of S is given by:
5
(a) π (f(x) − 9)2 − (g(x) − 9)2 dx 2
√
(b) 2π 5
5 2
(x+9)(f(x)−g(x))dx
(c) π (f(x) + 9)2 − (g(x) + 9)2 dx 2
3. In each option below is the description of a region R that has been rotated about a given axis to generate a solid of rotation S, and the integral that is used to calculate the volume S integrating on x.
Given also are some functions of y. Which of these functions of y are the integrand in the expression of the same solid of rotation when integrated on y?
(a) R is the region between y = x1/3 and y = x which is rotated about the x axis. The volume is
5 2
(x−9)(f(x)−g(x))dx (e) none of the above
(d) 2π
4
(x1/3)2 − 16 dx.
(b) R is the region between y = 2√x and y = x which is rotated about the y axis to form S. The
4 0
2 y4 The integrand using y: π y − 16
8 x2
given by: V = π
The integrand using y: 2π[y(4y − y3)]
0
volumeisgivenby: V =2π
√
x(2 x−x)dx.
√√
(c) Ristheregionbetweeny= 4−x2 andy=1onx∈[0, 3]whichisrotatedabouttheyaxis √
√
x( 4−x2−1)dx
(d) R is the region in the first quadrant between y = 6 and y = 2x + 2 which is rotated about the 2
3 0
toformS.Thevolumeisgivenby:V=2π The integrand using y is: π[(1 − (4 − y2)]
x(6−(2x+2))dx.
(e) none of the above
4. LettheregionRlieabovethecurvey=1−ax2 andbelowy=1on0≤x≤1. Youmayassume
0 < a ≤ 1 when making your sketch so that R lies entirely in the first quadrant.
Let S1 be the solid obtained by rotating R about the x axis and S2 be the solid obtained by rotating
R about the y axis.
Determine the value of the positive constant a such that the volumes of S1 and S2 are equal. The
value of a lies in which of the following intervals? (a) (0,1]
5
(b) (1,2] 55
(c) (2,3] 55
(d) (3,4] 55
(e) (4,1] 5
5. Consider the region R between y = f(x) and the x axis on [0,a] (a > 0). Use the shell method to calculate the volume of the solid obtained by rotating R around the y-axis given that dy = −xy and
yaxistoformS. Thevolumeisgivenby: V =2π The integrand using y is: π y − 12.
2
0
f(0) = 1. You may assume f(x) > 0.
(a) V = 2πa (b) V =2π−a (c) V = 2πf(a)
(d) V =2π(1−f(a)) (e) None of the above
6. Let f be positive with positive derivative on [0, 1]. Consider the region formed by rotating the region boundedbyy=f(0),y=f(x),x=0andx=1aboutthex-axis,whichhasvolumeV. Weconsider sums
π n
[f(i−1)2 −f(0)2] nn
i=1 π n
[f(i)2 −f(0)2] nn
i=1
Ln= Rn=
dx
Bn =2πf(1)−f(0)n f(0)+if(1)−f(0)1−f−1f(0)+if(1)−f(0) nnn
i=1
Un =2πf(1)−f(0)n f(0)+(i−1)f(1)−f(0)1−f−1f(0)+(i−1)f(1)−f(0). nnn
i=1 Choose all statements which are true.
(a) Each of Ln, Rn, Un, Bn converges to V . (b) EachLn
true.
(a) The population is always growing.
(b) The rate of population growth is fastest when M = 2P .
(c) The rate of population growth will never slow down.
(d) Eventually, the population will reach its maximum, i.e. at some time T.
(e) None of the above.
8. Which of the following are true:
(a) A first order differential equation of the form F (x, y, y′) = 0 must be linear (b) Ify=g(x)isanonzerosolutiontoy′ +2y=y2 thensoisy=2g(x)
(c) There are solutions to y′ = sin(x + y) that go through the origin (d) All solutions to y′ + ex = −y2 are decreasing
(e) none of the above
9. Consider the differential equation dy = y − x. Which of the following are true?
dx
(a) Any solution curve will have a local minimum along y = x.
(b) There exists a solution of the form y = ax + b, where a, b are constants.
(c) The solution curve passing through (0, 1) will also pass through (1, 2)
(d) Based on the direction field, a plausible value of an x intercept for the solution curve through (1, 1) is x = 2. (Do not attempt to find the solution to the DE. We are asking for an estimate only.)
(e) none of the above
10. Consider the differential equation modeling the harmonic oscillator
x′′(t) + k2x(t) = 0 (1) What is a correct solution to the equation?
(a) x = 2(sin(kt) + cos(kt)) (b) x = A cos(kt) + B sin(kt)
(c) x = A cos(kt) − B sin(kt) (d) x = k2 sin(t) cos(t)
(e) None of the above
11. Consider the following second order differential equation
x′′(t) + γx′(t) + ω2x(t) = 0, (2) where γ, ω are nonzero constants, and t ≥ 0, represents the time. We can define the total energy
of the system to be
E(t) = 1(x′(t))2 + 1ω2x(t)2. (3) 22
For t > 0, under what conditions is the total energy always decreasing?
(a) γ,ω2 >0 (b) γ,ω>0
(c) γ > 0, ω < 0 (d) ω > 0, γ ̸= 0 (e) ω < 0, γ ̸= 0