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Mathematics LibreTexts

Test 1(Mock Exam)

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These mock exams are provided to help you prepare for Term/Final tests. The best way to use these practice tests is to try the problems as if you were taking the test. Please don't look at the solution until you have attempted the question(s). Only reading through the answers or studying them, will typically not be helpful in preparing since it is too easy to convince yourself that you understand it.  

Exercise 1

Find the volume of the solid that results in when the region enclosed by x=y2 and 2y=2x is revolved about the line x=1.

Answer

7π15 unit3

Solution

Test1_1_PT.jpg

Point of intersections, we solve x=y2 and 2x=2y.

Now, y2=yy=0,1.

Method I:

Using Disk/Washer method,

V=π10((y+1)2(y2+1)2)dy=π10(2yy2y4)dy=7π15 unit3.

Method II:

Using Shell method, V=2π10(x+1)(xx)dx=2π10(x3/2x2+x1/2x)dx=2π(25x5/213x3+23x3/212x2)|10=7π15 unit3.

Exercise 2

Use cylindrical shells to find the volume generated when the region between the two curves y=x,y=0,x=1, and x=4 is revolved about the yaxis.

Answer

62π5 unit3.

Solution
V=2π41x(x)dx=2π41x32dx=4π5x52|41=62π5 unit3.

Exercise 3

Find the exact arc length of the curve x=18y4+14y2 from y=4 to y=9.

Answer

55167 unit

Solution

g(y)=12y312y3, therefore, 1+(g(y)2=1+(12y312y3)2=1+122y62(12y3)(12y3)+122y6=1+122y612+122y6=122y6+12+122y6=(12y3+12y3)2.

Hence Arc length=94(12y3+12y3)2dy=94(12y3+12y3)dy=(18y4+14y2)|94=55167 unit.

Exercise 4

Find the area of the surface generated by revolving 

y=ex+ex2,1x4,  about the x-axis.

Answer

π2((e8+6e8e2+e2))units

Solution

We have f(x)=12(exex), so [f(x)]2=(12(exex))2=14(e2x2+e2x).

Therefore, 1+[f(x)]2=1+14(e2x2+e2x)=14(4+e2x2+e2x)=14(e2x+2+e2x)=(12(ex+ex))2.

Surface area=ba2πf(x)1+[f(x)]2dx=412π(12(ex+ex))2dx=41π12(e2x+2+e2x)dx=π2(e2x2+2xe2x2)|41=π4((e8+16e8)(e2+4e2))=π4((e8+10e8e2+e2))units

Exercise 5

Calculate the following integrals:

  1. 3/21/2sin1x dx
  2. sin2(x)cos4(x)dx
  3. 2x22x2+50dx

 

Answer

3π6π12+132

x16164sin(4x)+148sin3(2x)+C

2x22x2+50 dx=225x2+25x+C.

Solution

1. Using integration by parts, let u=sin1(x) and dv=dx. Then du=11x2 and v=x. Therefore

3/21/2sin1x dx=xsin1(x)|3/21/23/21/2x1x2dx=(xsin1(x)+1x2)|3/21/2=((3/2)sin1(3/2)+13/2)((1/2)sin1(1/2)+11/2)=3π6π12+132.

2. 

sin2(x)cos4(x)dx=18(1cos(2x))(1+cos(2x))2dx=18(1cos2(2x))(1+cos(2x))dx=18sin2(2x)dx+18sin2(2x)cos(2x)dx=18(1cos(4x))dx+148sin3(2x)=x16164sin(4x)+148sin3(2x)+C

3. 

Let x=5tan(θ), then dx=5sec2(θ)dθ and x2+25=25sec2(θ).

2x22x2+50 dx=221x2x2+25dx=225sec2(θ)tan2(θ)sec(θ)dθ=225sec(θ)tan2(θ)dθ=225csc(θ)cot(θ)dθ=225csc(θ)+C.

Since tan(θ)=x5,sin(θ)=xx2+25.

t2q1.1.jpg

Therefore,

2x22x2+50 dx=225x2+25x+C.

 

Exercise 6

Determine if the following improper integral diverges or converges. If it converges, determine what number it converges to.

edxxln3(x)

Answer

The integral converges to 12.

Solution

edxxln3(x)=limttedxxln3(x)

Let u=ln(x), then du=1xdx. Then

edxxln3(x)=limttedxxln3(x)=limtx=tx=eduu3=limtx=tx=eu3du=limtu22|x=tx=e=limt12ln(x)2|x=tx=e=limt(12ln(t)2+12ln(e)2)=0+12=12.

Hence the integral converges to 12.


This page titled Test 1(Mock Exam) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.

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