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 this homework. I know it is a lot of questions but I would appreciate it if you can help me with as many as possible you can. Thanks.
![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%2F72ac8f78-2045-4eff-8be3-4c8859c13ea5%2Fzqzl32j_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]
![C. x = sint, y = sin 2t,
f. x = et ―t, y = 4et/2, [0,1]
6. A torus is formed by revolving the region bounded by the circle x² + y² = 4 around the line
x=3. Find the volume of this doughnut-shaped solid. [Hint: find the function that represents the
semicircle.]
7. A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of
1
2-x and the x-axis, around the x-axis, where x and y are measured in meters. Find
the tank's volume.
x²
8
1
8. An ornamental light bulb is designed by resolving the graph of y=x2-x/² on the interval
3
[0,1/3] around the x-axis, where x and y are measured in feet. Find the surface area of the bulb
and use the result to approximate the amount of glass needed to make the bulb (assume that
the glass is 0.015 inches thick).
9. Find the volume of the region bounded by y = ex, y = ex, x = 1 around the y-axis. (Use
shells.)
10. Differentiate.
a.
[0,1]
F(x) = [ - dt
t
b.
h(x) = f*t cos(t²) dt
So
c. y(x) = ex² e-t² dt
d. u(t)= ecos(sin² 3t)
e. q(t)
cos(Int)
t
f. v(t) = tan
=
3
a. y = 1 + 6x², [0,1]
b. y = (1-e-*), [0,2]
x3
c. y = ¹+1,[1,2]
4x
d.
3
y = x sinx, [0,2π]
g. g(x)=√t + sint dt
-x et
h. r(x) = f == dt
√√xt
i.a(t) = 2t-1
j. p(x)
k.s(t)
11. Find the length of arc in rectangular coordinates. You may need to calculate the value
numerically.
√x² + 3
arcsin(et + t)
e. y = In(secx), [0,7]
3
f. y = √√x, [0,1]
g. y = √x - x² + arcsin x, [0,1]
h. y = e-x²,[1,1]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa098e07d-b15e-4284-9177-08ea985b9919%2F72ac8f78-2045-4eff-8be3-4c8859c13ea5%2F59y7st_processed.jpeg&w=3840&q=75)
Transcribed Image Text:C. x = sint, y = sin 2t,
f. x = et ―t, y = 4et/2, [0,1]
6. A torus is formed by revolving the region bounded by the circle x² + y² = 4 around the line
x=3. Find the volume of this doughnut-shaped solid. [Hint: find the function that represents the
semicircle.]
7. A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of
1
2-x and the x-axis, around the x-axis, where x and y are measured in meters. Find
the tank's volume.
x²
8
1
8. An ornamental light bulb is designed by resolving the graph of y=x2-x/² on the interval
3
[0,1/3] around the x-axis, where x and y are measured in feet. Find the surface area of the bulb
and use the result to approximate the amount of glass needed to make the bulb (assume that
the glass is 0.015 inches thick).
9. Find the volume of the region bounded by y = ex, y = ex, x = 1 around the y-axis. (Use
shells.)
10. Differentiate.
a.
[0,1]
F(x) = [ - dt
t
b.
h(x) = f*t cos(t²) dt
So
c. y(x) = ex² e-t² dt
d. u(t)= ecos(sin² 3t)
e. q(t)
cos(Int)
t
f. v(t) = tan
=
3
a. y = 1 + 6x², [0,1]
b. y = (1-e-*), [0,2]
x3
c. y = ¹+1,[1,2]
4x
d.
3
y = x sinx, [0,2π]
g. g(x)=√t + sint dt
-x et
h. r(x) = f == dt
√√xt
i.a(t) = 2t-1
j. p(x)
k.s(t)
11. Find the length of arc in rectangular coordinates. You may need to calculate the value
numerically.
√x² + 3
arcsin(et + t)
e. y = In(secx), [0,7]
3
f. y = √√x, [0,1]
g. y = √x - x² + arcsin x, [0,1]
h. y = e-x²,[1,1]
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