CALC 2 Stetson University Volume of The Cap of The Sphere of Radius R Questions

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Stetson University Summer Term 2020 Calculus II Exam #1 10 questions – Due Monday 6/15 by 11:59pm (submit to BB journals) Name: This work is to be done on your own – and without the help of another person (other than me). You may use resources online, from your text, webassign, etc. By signing this agreement to attest to this work being exclusively yours. Signature:____________________ #1. Find the area of the shaded region. #2. Derive the volume of a sphere by revolving the semi-circle defined by y = r 2 βˆ’ x 2 about the x-axis where r is the (fixed) radius if the semicircle. #3. Find the volume of the cap of the sphere of radius r. The height of the cap is h. (See picture). #4. Use the method of cylindrical shells to find the volume of the solid torus indicated in the figure below. #5. Evaluate e a sin(b )d where a, b οƒŽ #6. Evaluate the integral .  ln(ax + b)dx where a, b οƒŽ . #7. Evaluate the integral  sin(ln x)dx #8. Evaluate the integral x 2 cos(mx)dx where m οƒŽ + . #9. Evaluate r 2 ln(r )dr #10. Evaluate the integral
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Explanation & Answer

Hi, here are the answers. If you do not have any other questions, please mark this session as complete. Thank you! πŸ˜‡

1.

Find the area of the shaded region.
The shaded region can be divided into
rectangles along the y-axis. The area is the
integral of the difference between the right and
left functions.
7
2

π΄π‘Ÿπ‘’π‘”π‘–π‘œπ‘› = ∫ [(3𝑦 βˆ’ 𝑦 2 ) βˆ’ (𝑦 2 βˆ’ 4𝑦)]𝑑𝑦
0

7
2

π΄π‘Ÿπ‘’π‘”π‘–π‘œπ‘› = ∫ (βˆ’2𝑦 2 + 7𝑦)𝑑𝑦
0

π΄π‘Ÿπ‘’π‘”π‘–π‘œπ‘›
π΄π‘Ÿπ‘’π‘”π‘–π‘œπ‘›
π΄π‘Ÿπ‘’π‘”π‘–π‘œπ‘›
π΄π‘Ÿπ‘’π‘”π‘–π‘œπ‘›

7

2𝑦 3 7𝑦 2 2
= [βˆ’
+
]
3
2
0
7 3
7 2
2( )
7( )
2(0)3 7(0)2
= βˆ’ 2 + 2 βˆ’ (βˆ’
+
)
3
2
3
2
2(343
7(49
)
8 )
=βˆ’
+ 4
3
2
343
=
π‘ π‘žπ‘’π‘Žπ‘Ÿπ‘’ 𝑒𝑛𝑖𝑑𝑠
24

2.

Derive the volume of a sphere by revolving a semi-circle defined by 𝑦 = βˆšπ‘Ÿ 2 βˆ’ π‘₯ 2 about the
x-axis where r is the (fixed) radius of the semi-circle.

Divided the sphere into cylindrical elements along the y-axis with radius π‘Ÿ and altitude 𝑑𝑦. The
volume of this cylindrical element is 𝑑𝑉 = πœ‹π‘₯ 2 𝑑𝑦. The sum of the cylindrical elements from 0 to r
π‘Ÿ
is a hemisphere, so twice this value gives the sphere volume, or π‘‰π‘ π‘β„Žπ‘’π‘Ÿπ‘’ = 2(∫0 πœ‹π‘₯ 2 𝑑𝑦). Now,
we get the value of x in terms of y and substitute this to the equation for the volume.
𝑦 = βˆšπ‘Ÿ 2 βˆ’ π‘₯ 2
π‘₯2 = π‘Ÿ2 βˆ’ 𝑦2
π‘Ÿ

π‘‰π‘ π‘β„Žπ‘’π‘Ÿπ‘’ = 2 (∫ πœ‹(π‘Ÿ 2 βˆ’ 𝑦 2 )𝑑𝑦)
0
π‘Ÿ

π‘‰π‘ π‘β„Žπ‘’π‘Ÿπ‘’ = 2πœ‹ (∫ (π‘Ÿ 2 βˆ’ 𝑦 2 )𝑑𝑦)
0

π‘‰π‘ π‘β„Žπ‘’π‘Ÿπ‘’

𝑦3 π‘Ÿ
= 2πœ‹ [π‘Ÿ 𝑦 βˆ’ ]
3 0
2

π‘‰π‘ π‘β„Žπ‘’π‘Ÿπ‘’ = 2πœ‹ [((π‘Ÿ 2 )(π‘Ÿ) βˆ’
π‘‰π‘ π‘β„Žπ‘’π‘Ÿπ‘’ = 2πœ‹ [π‘Ÿ 3 βˆ’
2π‘Ÿ 3
π‘‰π‘ π‘β„Žπ‘’π‘Ÿπ‘’ = 2πœ‹ [
]
3
4πœ‹π‘Ÿ 3
π‘‰π‘ π‘β„Žπ‘’π‘Ÿπ‘’ =
3

π‘Ÿ3
]
3

(π‘Ÿ)3
(0)3
) βˆ’ ((π‘Ÿ 2 )(0) βˆ’
)]
3
3

3.

Find the volume of the cap of the sphere of radius r. The height of the cap is h.

The volume of the cap of the sphere is the sum of the cylindrical elements (with radius x) from the
base of the cap at 𝑦 = π‘Ÿ βˆ’ β„Ž to the top at οΏ½...


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