Question 2 (40 points): Consider the region R bounded by y = x² and y=x. (a) (10 points) Sketch the region R and find the area of the region R. (b) (15 points) Find the volume of the solid obtained by revolving the region R about the line y = 1 using disk-washer method. (c) (15 points) Find the volume obtained by revolving the region R about the x = 1 using cylindrical-shell method. Solution: (a) (0.0) (1.1) Find the intersection points: (0,0) and (1,1). Area of R 04 " 1 LE 5 points = √ ²( x − x² ) d x = ( ² ² = ²) | = ¦¦- (b) 2 3 16 Take a vertical slice at x with thickness dx and rotate it about the y = 1. Then volume of the washer is dV = [(1 − x²)² - π(1-x)²]dx = n(x² - 3x² + 2x)dx V = √ dV = √(x² -3x² + 2x)dx 5 points ------- 10 points - 4 points =(-1+1)= (c) Take a vertical slice at x with thickness dx, rotate it about the line x = 1 point x= 1. It forms a cylindrical shell. Its volume is given by dV = 2(1-x)(x − x²)dx v=fdv=2(1-x)(x - x²)dx 10 points
Question 2 (40 points): Consider the region R bounded by y = x² and y=x. (a) (10 points) Sketch the region R and find the area of the region R. (b) (15 points) Find the volume of the solid obtained by revolving the region R about the line y = 1 using disk-washer method. (c) (15 points) Find the volume obtained by revolving the region R about the x = 1 using cylindrical-shell method. Solution: (a) (0.0) (1.1) Find the intersection points: (0,0) and (1,1). Area of R 04 " 1 LE 5 points = √ ²( x − x² ) d x = ( ² ² = ²) | = ¦¦- (b) 2 3 16 Take a vertical slice at x with thickness dx and rotate it about the y = 1. Then volume of the washer is dV = [(1 − x²)² - π(1-x)²]dx = n(x² - 3x² + 2x)dx V = √ dV = √(x² -3x² + 2x)dx 5 points ------- 10 points - 4 points =(-1+1)= (c) Take a vertical slice at x with thickness dx, rotate it about the line x = 1 point x= 1. It forms a cylindrical shell. Its volume is given by dV = 2(1-x)(x − x²)dx v=fdv=2(1-x)(x - x²)dx 10 points
Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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I am studying for a test I have tomorrow, this is one of the example problems that will be one the test, can you show the equations used in this problem answer and then explain. Can you also show how to make the graph and how changing something in the question will effected it as an example.
![Question 2 (40 points): Consider the region R bounded by y = x² and y=x.
(a) (10 points) Sketch the region R and find the area of the region R.
(b) (15 points) Find the volume of the solid obtained by revolving the region R about
the line y = 1 using disk-washer method.
(c) (15 points) Find the volume obtained by revolving the region R about the x = 1
using cylindrical-shell method.
Solution:
(a)
(0.0)
(1.1)
Find the intersection points: (0,0) and (1,1).
Area of R
04
"
1
LE
5 points
= √ ²( x − x² ) d x = ( ² ² = ²) | = ¦¦-
(b)
2
3
16
Take a vertical slice at x with thickness dx and rotate it about the y = 1.
Then volume of the washer is
dV = [(1 − x²)² - π(1-x)²]dx = n(x² - 3x² + 2x)dx
V = √ dV = √(x² -3x² + 2x)dx
5 points
------- 10 points
-
4 points
=(-1+1)=
(c) Take a vertical slice at x with thickness dx, rotate it about the line x =
1 point
x= 1.
It forms a cylindrical shell. Its volume is given by
dV = 2(1-x)(x − x²)dx
v=fdv=2(1-x)(x - x²)dx
10 points](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6356ceb6-8738-457b-86a1-1decb02218d2%2Fc1f53ff5-2345-45ef-9346-cdc1ab3b891a%2Fhi3gk9_processed.png&w=3840&q=75)
Transcribed Image Text:Question 2 (40 points): Consider the region R bounded by y = x² and y=x.
(a) (10 points) Sketch the region R and find the area of the region R.
(b) (15 points) Find the volume of the solid obtained by revolving the region R about
the line y = 1 using disk-washer method.
(c) (15 points) Find the volume obtained by revolving the region R about the x = 1
using cylindrical-shell method.
Solution:
(a)
(0.0)
(1.1)
Find the intersection points: (0,0) and (1,1).
Area of R
04
"
1
LE
5 points
= √ ²( x − x² ) d x = ( ² ² = ²) | = ¦¦-
(b)
2
3
16
Take a vertical slice at x with thickness dx and rotate it about the y = 1.
Then volume of the washer is
dV = [(1 − x²)² - π(1-x)²]dx = n(x² - 3x² + 2x)dx
V = √ dV = √(x² -3x² + 2x)dx
5 points
------- 10 points
-
4 points
=(-1+1)=
(c) Take a vertical slice at x with thickness dx, rotate it about the line x =
1 point
x= 1.
It forms a cylindrical shell. Its volume is given by
dV = 2(1-x)(x − x²)dx
v=fdv=2(1-x)(x - x²)dx
10 points
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