The region bounded by the curve y = x² + 1 and the line y-x+3 is revolved about the x-axis to generate a solid. Find the volume of this solid. Solution:
The region bounded by the curve y = x² + 1 and the line y-x+3 is revolved about the x-axis to generate a solid. Find the volume of this solid. Solution:
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
![The region bounded by the curve y = x² + 1 and the line y-x+3 is
revolved about the x-axis to generate a solid. Find the volume of this solid.
Solution:
x² + 1 = x + 3
x²+x-2=0
(x+2)(x-1) = 0
x=-2, x = 1
Outer radius: R(x) = x+3
Inner radius: r(x) = x² + 1
= L" = ([R(x)]² - [r{(x)]³) dx
V =
(-2,5)
R(x)--x+3
y--x+3
r(x)=x²+1
(1,2)
(-x+3)² - (x² + 1)²) dx
= r(=x
-2
= #(8
6x - x²- x²) dx
Interval of
integration
(a)
(-2.5)
ROX)
-x+3
(1, 2)
+1
=π 8x
3x²------- 117
Washer cross section
=
Outer radius: R(x)--x+3
Inner radius: r(x) = x² + 1
(b)
H.W. 1:
Solve the previous example by rotating the region about the
line y = -1 ?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0baaf4e4-5654-4278-a18e-0b3d1e9bbff7%2F09d96d2c-6ff6-441d-9f9d-cae80185d8d9%2Fdq3owlw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The region bounded by the curve y = x² + 1 and the line y-x+3 is
revolved about the x-axis to generate a solid. Find the volume of this solid.
Solution:
x² + 1 = x + 3
x²+x-2=0
(x+2)(x-1) = 0
x=-2, x = 1
Outer radius: R(x) = x+3
Inner radius: r(x) = x² + 1
= L" = ([R(x)]² - [r{(x)]³) dx
V =
(-2,5)
R(x)--x+3
y--x+3
r(x)=x²+1
(1,2)
(-x+3)² - (x² + 1)²) dx
= r(=x
-2
= #(8
6x - x²- x²) dx
Interval of
integration
(a)
(-2.5)
ROX)
-x+3
(1, 2)
+1
=π 8x
3x²------- 117
Washer cross section
=
Outer radius: R(x)--x+3
Inner radius: r(x) = x² + 1
(b)
H.W. 1:
Solve the previous example by rotating the region about the
line y = -1 ?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Introduce to the given information
VIEWStep 2: Determining the intersection points of the curve
VIEWStep 3: Sketching the graph of the region and the solid
VIEWStep 4: Determining the expression for outer radius, inner radius and the limits for the integration
VIEWStep 5: Determining the volume of the solid
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