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Calculus I Test Iv

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Calculus
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UCF
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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:
.

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#1. Find the area of the shaded region
Solution:-
Let an small dy at a distance of y from x-axis as shown in the figure.
Area of shaded part is given by
=
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 
 


=

 


=

= 14.29

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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 Solution:- Let an small dy at a distance of y from x-axis as shown in the figure. Area of shaded part is given by 7/2 = ∫0 (3𝑦 − 𝑦 2 − 𝑦 2 + 4𝑦)𝑑𝑦 7/2 = ∫0 (−2𝑦 2 + 7𝑦)𝑑𝑦 7 2 = 2 𝑦 2 − 3 𝑦 3 | 7/2 0 = 14.29 #2. Derive the volume of a sphere by revolving the semi-circle defined by about the x-axis where r is the (fixed) radius if the semiy= circle. Solution : x Take a disc of width dx at a distance of x from y-axis radius of disc will be y =√𝑟 2 − 𝑥 2 area of disc = 𝜋 × 𝑟𝑎𝑑𝑖𝑢𝑠^2 = 𝜋 (𝑟 2 − 𝑥 2 ) So, dv = 𝜋 (𝑟 2 − 𝑥 2 )𝑑𝑥 𝑟 V = ∫−𝑟 𝜋 (𝑟 2 − 𝑥 2 )𝑑𝑥 𝑟 = 2 ∫0 𝜋 (𝑟 2 − 𝑥 2 )𝑑𝑥 = 2𝜋(𝑟 2 𝑥 − =4𝜋 𝑟3 𝑥3 3 ) | 𝑟0 3 #3. Find the volume of the cap of the sphere of radius r. The height of the cap is h. (See picture). Solution: √𝑟 2 − 𝑦 2 y r Take a disc of width dy at a distance of y from x axis Area of disc = 𝜋 × 𝑟𝑎𝑑𝑖𝑢𝑠^2 = 𝜋 (𝑟 2 − 𝑦 2 ) ...
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