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
-
Point of intersections, we solve x=y2 and 2x=2y.
Now, y2=y⟹y=0,1.
Method I:
Using Disk/Washer method,
V=π∫10((y+1)2−(y2+1)2)dy=π∫10(2y−y2−y4)dy=7π15 unit3.
Method II:
Using Shell method, V=2π∫10(x+1)(√x−x)dx=2π∫10(x3/2−x2+x1/2−x)dx=2π(25x5/2−13x3+23x3/2−12x2)|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 y−axis.
- Answer
-
62π5 unit3.
- Solution
- V=2π∫41x(√x)dx=2π∫41x32dx=4π5x52|41=62π5 unit3.
Exercise 3
Find the exact arc length of the curve x=18y4+14y−2 from y=4 to y=9.
- Answer
-
55167 unit
- Solution
-
g′(y)=12y3−12y−3, therefore, 1+(g′(y)2=1+(12y3−12y−3)2=1+122y6−2(12y3)(12y−3)+122y−6=1+122y6−12+122y−6=122y6+12+122y−6=(12y3+12y−3)2.
Hence Arc length=∫94√(12y3+12y−3)2dy=∫94(12y3+12y−3)dy=(18y4+−14y−2)|94=55167 unit.
Exercise 4
Find the area of the surface generated by revolving
y=ex+e−x2,1≤x≤4, about the x-axis.
- Answer
-
π2((e8+6−e−8−e2+e−2))units
- Solution
-
We have f′(x)=12(ex−e−x), so [f′(x)]2=(12(ex−e−x))2=14(e2x−2+e−2x).
Therefore, 1+[f′(x)]2=1+14(e2x−2+e−2x)=14(4+e2x−2+e−2x)=14(e2x+2+e−2x)=(12(ex+e−x))2.
Surface area=∫ba2πf(x)√1+[f′(x)]2dx=∫412π(12(ex+e−x))2dx=∫41π12(e2x+2+e−2x)dx=π2(e2x2+2x−e−2x2)|41=π4((e8+16−e−8)−(e2+4−e−2))=π4((e8+10−e−8−e2+e−2))units
Exercise 5
Calculate the following integrals:
- ∫√3/21/2sin−1x dx
- ∫sin2(x)cos4(x)dx
- ∫2x2√2x2+50dx
- Answer
-
√3π6−π12+1−√32
x16−164sin(4x)+148sin3(2x)+C
∫2x2√2x2+50 dx=−√225√x2+25x+C.
- Solution
-
1. Using integration by parts, let u=sin−1(x) and dv=dx. Then du=1√1−x2 and v=x. Therefore
∫√3/21/2sin−1x dx=xsin−1(x)|√3/21/2−∫√3/21/2x√1−x2dx=(xsin−1(x)+√1−x2)|√3/21/2=((√3/2)sin−1(√3/2)+√1−√3/2)−((1/2)sin−1(1/2)+√1−1/2)=√3π6−π12+1−√32.
2.
∫sin2(x)cos4(x)dx=18∫(1−cos(2x))(1+cos(2x))2dx=18∫(1−cos2(2x))(1+cos(2x))dx=18∫sin2(2x)dx+18∫sin2(2x)cos(2x)dx=18∫(1−cos(4x))dx+148sin3(2x)=x16−164sin(4x)+148sin3(2x)+C
3.
Let x=5tan(θ), then dx=5sec2(θ)dθ and x2+25=25sec2(θ).
∫2x2√2x2+50 dx=2√2∫1x2√x2+25dx=√225∫sec2(θ)tan2(θ)sec(θ)dθ=√225∫sec(θ)tan2(θ)dθ=√225∫csc(θ)cot(θ)dθ=−√225csc(θ)+C.
Since tan(θ)=x5,sin(θ)=x√x2+25.
Therefore,
∫2x2√2x2+50 dx=−√225√x2+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)=limt→∞∫tedxxln3(x)
Let u=ln(x), then du=1xdx. Then
∫∞edxxln3(x)=limt→∞∫tedxxln3(x)=limt→∞∫x=tx=eduu3=limt→∞∫x=tx=eu−3du=limt→∞u−2−2|x=tx=e=limt→∞−12ln(x)2|x=tx=e=limt→∞(−12ln(t)2+12ln(e)2)=0+12=12.
Hence the integral converges to 12.