Math 2204 Test 2 Practice Test Version A
.pdf
School
Indus International School, Bangalore**We aren't endorsed by this school
Course
AC 1504
Subject
Mathematics
Date
Dec 17, 2024
Pages
10
Uploaded by ChefSandpiperPerson697
MATH 2204 Test 2 Version ANAME:CRN:Honor Pledge:I have neither given nor received aid on this exam. Signature:1
Offcial Math 2204 Scratch Paper2
Offcial Math 2204 Scratch Paper Cont.3
Multiple Choice•No partial credit will be given.•Clearly circleyour answer.•No calculator!1. Which of the following equations in spherical coordinates represents the surface given by(x2+y2+z2−z)2=x2+y2+z2?Circle only one answer.(A)ρ2= sin(ϕ)−cos(ϕ)(B)ρ= 1−cos(ϕ)(C)ρ= sin(θ) cos(ϕ)(D)ρ= 1−cos(θ) sin(ϕ)2. Which of the following represents the area of(x2−x+y2)2=x2+y2? Circle only one answer.(A)∫2π0∫1+cos(θ)0rdrdθ(B)∫π2−π2∫1+cos(θ)0rdrdθ(C)∫2π0∫(r2−rcos(θ))2rrdrdθ(D)∫π2−π2∫(r2−rcos(θ))2rrdrdθ4
3. Which of the following integrals is equal to∫10∫√1−x20∫√x2+y2x2+y2dzdydxin cylindrical coordinates written indθdrdzorder? Circle only one answer.(A)∫π20∫10∫rr2rdθdrdz(B)∫π20∫10∫r2rrdθdrdz(C)∫10∫√zz∫π20rdθdrdz(D)∫10∫√z0∫π20rdθdrdz4. Which of the following integrals is equal to∫10∫√1−x20∫√x2+y2x2+y2dzdydxin spherical coordinates? Circle only one answer.(A)∫π20∫π2π4∫cot(ϕ) csc(ϕ)0ρ2sin(ϕ)dρdϕdθ(B)∫π20∫π2π4∫ρsin(ϕ)ρ2sin2(ϕ)ρ2sin(ϕ)dρdϕdθ(C)∫π2π4∫π20∫cot(ϕ) csc(ϕ)0ρ2sin(ϕ)dρdϕdθ(D)∫π2π4∫π20∫ρsin(ϕ)ρ2sin2(ϕ)ρ2sin(ϕ)dρdϕdθ5
Free Response•Show reasoning that is complete and correct by the standards of this course.•Whenever using theorems, you should explicitly check that all hypotheses are satisfied.•Improper use of (or the absence of) proper notation will be penalized.•No calculator!5. Identify the surface given byρ=−2 sin(ϕ) cos(θ).6. Evaluate the integral off(x, y, z) =x2+y2+z2in the region bounded between the planex=−1and the surfaceρ=−2 sin(ϕ) cos(θ)using spherical coordinates.6
7. Evaluate the integral off(x, y, z) =x2+y2+z2in the region bounded between the planex=−1and the surfaceρ=−2 sin(ϕ) cos(θ)using cylindrical coordinates.7
8. LetEbe the region bounded by the cylindersx2+y2= 1andx2+z2= 1.(a) Sketch the region.(b) Write the volume as an integral in rectangular coordinates.(c) Write the volume as an integral in cylindrical coordinates.8
(d) Evaluate both integrals and show that the volumes are equivalent.9
(Challenge) LetEinstead be the region bounded by the cylindersx2+y2= 1,x2+z2= 1, andy2+z2= 1. Find thevolume of the region.hi10