4. Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space R". [Hint: The formulas are different for n even and n odd.]
4. Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space R". [Hint: The formulas are different for n even and n odd.]
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter61: Areas Of Circles, Sectors, And Segments
Section: Chapter Questions
Problem 50A
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Can you please solve q4? Thanks
Also, provide detailed answer.
![In this project we find formulas for the volume enclosed by a hypersphere in n-dimensional
space.
1. Use a double integral and trigonometric substitution, together with the formula
1
1
Scos
u du =-u+÷sin(2u)+C from the Table of Integrals, to find the area of a circle with radius r.
4
2. Use a triple integral and trigonometric substitution to find the volume of a sphere with radius r.
3. Use a quadruple integral to find the hypervolume enclosed by the hypersphere in
x² + y² +z? + w² = r? in R* . (Use only trigonometric substitution and the reduction formulas for
Ssin" x- dx or fcos" x•dx .)
4. Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space
R". [Hint: The formulas are different for n even and n odd.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ee2210f-6539-44d4-8e38-60565143d3fa%2F36d82208-ab72-4988-8cd1-c127d7aebb5c%2Fopn069_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In this project we find formulas for the volume enclosed by a hypersphere in n-dimensional
space.
1. Use a double integral and trigonometric substitution, together with the formula
1
1
Scos
u du =-u+÷sin(2u)+C from the Table of Integrals, to find the area of a circle with radius r.
4
2. Use a triple integral and trigonometric substitution to find the volume of a sphere with radius r.
3. Use a quadruple integral to find the hypervolume enclosed by the hypersphere in
x² + y² +z? + w² = r? in R* . (Use only trigonometric substitution and the reduction formulas for
Ssin" x- dx or fcos" x•dx .)
4. Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space
R". [Hint: The formulas are different for n even and n odd.]
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Solve all please
![1. Use a double integral and trigonometric substitution, together with the formula
1
1
[cos²u du ==u+
u+ sin(2u)+C from the Table of Integrals, to find the area of a circle with radius r.
2
2. Use a triple integral and trigonometric substitution to find the volume of a sphere with radius r.](https://content.bartleby.com/qna-images/question/62d7bd71-12a8-4d88-8c42-1257ba6dd969/d275834b-4311-47aa-8552-4b5c252f7638/v3fdhw9_thumbnail.png)
Transcribed Image Text:1. Use a double integral and trigonometric substitution, together with the formula
1
1
[cos²u du ==u+
u+ sin(2u)+C from the Table of Integrals, to find the area of a circle with radius r.
2
2. Use a triple integral and trigonometric substitution to find the volume of a sphere with radius r.
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