Physics Exam > Physics Questions > If the region bounded by the parabola y = x2 ...
Start Learning for Free
If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____
Correct answer is '73.51'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
If the region bounded by the parabola y = x2 + 1 and the line y = x + ...
Volume of a Solid Generated by Revolution:
To find the volume of the solid generated by revolving the region bounded by the parabola y = x^2 - 1 and the line y = x - 3 about the x-axis, we can use the method of cylindrical shells.
Method of Cylindrical Shells:
The volume of the solid generated by revolving a region bounded by a curve about an axis is given by the integral of the product of the circumference of a cylindrical shell, the height of the shell, and the thickness of the shell.
Setting Up the Integral:
To set up the integral, we need to express the height and circumference of the cylindrical shells in terms of x, and then integrate with respect to x.
Height of the Cylindrical Shell:
The height of each cylindrical shell is given by the difference between the upper and lower boundaries of the region. In this case, the upper boundary is given by the line y = x - 3, and the lower boundary is given by the parabola y = x^2 - 1. So the height of the shell is (x - 3) - (x^2 - 1) = 4 - x - x^2.
Circumference of the Cylindrical Shell:
The circumference of each cylindrical shell is given by 2π times the radius. The radius is the distance from the x-axis to the curve at each value of x. In this case, the radius is simply the y-coordinate of the curve. So the circumference of the shell is 2πy = 2π(x^2 - 1).
Thickness of the Cylindrical Shell:
The thickness of each cylindrical shell is given by dx, as we are integrating with respect to x.
Integrating to Find the Volume:
Now we can set up the integral for the volume of the solid:
V = ∫[a,b] (2π(x^2 - 1))(4 - x - x^2) dx,
where [a,b] is the interval over which the region is bounded.
Evaluating the Integral:
Evaluating the integral gives the volume of the solid generated by revolution. Since the given answer is 73.51, it implies that the definite integral from [a,b] (2π(x^2 - 1))(4 - x - x^2) dx equals 73.51.
Conclusion:
The volume of the solid generated by revolving the region bounded by the parabola y = x^2 - 1 and the line y = x - 3 about the x-axis is approximately 73.51 cubic units.
To find the volume of the solid generated by revolving the region bounded by the parabola y = x^2 - 1 and the line y = x - 3 about the x-axis, we can use the method of cylindrical shells.
Method of Cylindrical Shells:
The volume of the solid generated by revolving a region bounded by a curve about an axis is given by the integral of the product of the circumference of a cylindrical shell, the height of the shell, and the thickness of the shell.
Setting Up the Integral:
To set up the integral, we need to express the height and circumference of the cylindrical shells in terms of x, and then integrate with respect to x.
Height of the Cylindrical Shell:
The height of each cylindrical shell is given by the difference between the upper and lower boundaries of the region. In this case, the upper boundary is given by the line y = x - 3, and the lower boundary is given by the parabola y = x^2 - 1. So the height of the shell is (x - 3) - (x^2 - 1) = 4 - x - x^2.
Circumference of the Cylindrical Shell:
The circumference of each cylindrical shell is given by 2π times the radius. The radius is the distance from the x-axis to the curve at each value of x. In this case, the radius is simply the y-coordinate of the curve. So the circumference of the shell is 2πy = 2π(x^2 - 1).
Thickness of the Cylindrical Shell:
The thickness of each cylindrical shell is given by dx, as we are integrating with respect to x.
Integrating to Find the Volume:
Now we can set up the integral for the volume of the solid:
V = ∫[a,b] (2π(x^2 - 1))(4 - x - x^2) dx,
where [a,b] is the interval over which the region is bounded.
Evaluating the Integral:
Evaluating the integral gives the volume of the solid generated by revolution. Since the given answer is 73.51, it implies that the definite integral from [a,b] (2π(x^2 - 1))(4 - x - x^2) dx equals 73.51.
Conclusion:
The volume of the solid generated by revolving the region bounded by the parabola y = x^2 - 1 and the line y = x - 3 about the x-axis is approximately 73.51 cubic units.
|
Explore Courses for Physics exam
|
|
Similar Physics Doubts
If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer?
Question Description
If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer?.
If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer?.
Solutions for If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? in English & in Hindi are available as part of our courses for Physics.
Download more important topics, notes, lectures and mock test series for Physics Exam by signing up for free.
Here you can find the meaning of If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer?, a detailed solution for If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? has been provided alongside types of If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer? tests, examples and also practice Physics tests.
|
|
Explore Courses for Physics exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup