Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
I need help with number 4.
![MTH 264, Homework #1, Summer 2022 Name
Instructions: Write your work up neatly and attach to this page. Record your final answers (only)
directly on this page if they are short; if too long indicate which page of the work the answer is on and
mark it clearly. Use exact values unless specifically asked to round.
1.
Find the area bounded by the curves and sketch the region.
a. f(x)=x² - 4x² and g(x)=x² - 4x
b.
y = x² - 2x, y = x +4
c. y = cos x, y = 2
d. y = x³, y = x
e. y = |x|, y = x² - 2
cos x, [0, 2π]
2. Set up the integral to find the value of the area bounded by the graphs. You do not need to
evaluate it, but do sketch the region.
a.
y =
2
1+x4
‚y x²
=
b. y = cos x, y = x + 2 sin4 x
i.
ii.
y-axis
the line y = 8
3. Calculate the volumes of the solids of revolution as indicated below. Calculate the volume both
with the shell method and with the disk or washer method and verify that the volume is the
same in each case.
a. y=2x², y=0, x=2 Revolved around the:
10
b. y=₁y=0,₂x = 1, x = 5 Revolved around:
i.
y-axis
y = 10
ii.
c. y = ex, y = 1, x = 2 Revolved around:
i.
y-axis
x-axis
ii.
d. x = y², x = 1 - y² Revolved around:
i.
y = 1
e. y = x³, y = 0, x = 1 Revolved around:
i.
ii.
y-axis
x-axis
4. Find the surface area when the function y
=
x³
1.3
f. y = ex, y = x² − 1, x = −1, x = 1
g. x = 1 - y², x = y² - 1
h. x = y¹, y = √2x, y = 0
1
X
i.y ==, y = x, y = x, x > 0
j. y = x²e-x, y = xe-x
c. y = e¹-x²
1
, y=x4
+
6 2x
iii. x-axis
iv. the line x = 2
iii. x-axis
iv. x = 5
iii. y = 0
iv. x = 3
ii. x = 3
iii. x = 1
iv. y = 1
on the interval 1≤x≤2.
5. Find the area of the surface generated by revolving the curve about i) the x-axis, ii) the y-axis.
Sketch the graph. You may need to integrate numerically.
a. x=t, y=4-2t [0,4]
d. x = a cos 0, y = asin0 [0,2π]
b. x = t³, y = t², [0,1]
e. x = t cost, y = t sint, [0,1]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa098e07d-b15e-4284-9177-08ea985b9919%2F78914c18-d03d-42b5-ba89-adc154548dad%2Fb8cccot_processed.jpeg&w=3840&q=75)
Transcribed Image Text:MTH 264, Homework #1, Summer 2022 Name
Instructions: Write your work up neatly and attach to this page. Record your final answers (only)
directly on this page if they are short; if too long indicate which page of the work the answer is on and
mark it clearly. Use exact values unless specifically asked to round.
1.
Find the area bounded by the curves and sketch the region.
a. f(x)=x² - 4x² and g(x)=x² - 4x
b.
y = x² - 2x, y = x +4
c. y = cos x, y = 2
d. y = x³, y = x
e. y = |x|, y = x² - 2
cos x, [0, 2π]
2. Set up the integral to find the value of the area bounded by the graphs. You do not need to
evaluate it, but do sketch the region.
a.
y =
2
1+x4
‚y x²
=
b. y = cos x, y = x + 2 sin4 x
i.
ii.
y-axis
the line y = 8
3. Calculate the volumes of the solids of revolution as indicated below. Calculate the volume both
with the shell method and with the disk or washer method and verify that the volume is the
same in each case.
a. y=2x², y=0, x=2 Revolved around the:
10
b. y=₁y=0,₂x = 1, x = 5 Revolved around:
i.
y-axis
y = 10
ii.
c. y = ex, y = 1, x = 2 Revolved around:
i.
y-axis
x-axis
ii.
d. x = y², x = 1 - y² Revolved around:
i.
y = 1
e. y = x³, y = 0, x = 1 Revolved around:
i.
ii.
y-axis
x-axis
4. Find the surface area when the function y
=
x³
1.3
f. y = ex, y = x² − 1, x = −1, x = 1
g. x = 1 - y², x = y² - 1
h. x = y¹, y = √2x, y = 0
1
X
i.y ==, y = x, y = x, x > 0
j. y = x²e-x, y = xe-x
c. y = e¹-x²
1
, y=x4
+
6 2x
iii. x-axis
iv. the line x = 2
iii. x-axis
iv. x = 5
iii. y = 0
iv. x = 3
ii. x = 3
iii. x = 1
iv. y = 1
on the interval 1≤x≤2.
5. Find the area of the surface generated by revolving the curve about i) the x-axis, ii) the y-axis.
Sketch the graph. You may need to integrate numerically.
a. x=t, y=4-2t [0,4]
d. x = a cos 0, y = asin0 [0,2π]
b. x = t³, y = t², [0,1]
e. x = t cost, y = t sint, [0,1]
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