1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Elina [12.6K]
3 years ago
13

Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving

the plane region about the y-axis. y
Mathematics
1 answer:
faust18 [17]3 years ago
3 0

Answer:

Volume = \frac{384}{7}\pi

Step-by-step explanation:

Given (Missing Information):

y = x^\frac{3}{2}; y = 8; x=0

Required

Determine the volume

Using Shell Method:

V = 2\pi \int\limits^a_b {p(y)h(y)} \, dy

First solve for a and b.

y = x^\frac{3}{2} and y = 8

Substitute 8 for y

8 = x^\frac{3}{2}

Take 2/3 root of both sides

8^\frac{2}{3} = x^{\frac{3}{2}*\frac{2}{3}}

8^\frac{2}{3} = x

2^{3*\frac{2}{3}} = x

2^2 = x

4 =x

x = 4

This implies that:

a = 4

For x=0

This implies that:

b=0

So, we have:

V = 2\pi \int\limits^a_b {p(y)h(y)} \, dy

V = 2\pi \int\limits^4_0 {p(y)h(y)} \, dy

The volume of the solid becomes:

V = 2\pi \int\limits^4_0 {x(8 - x^{\frac{3}{2}}}) \, dx

Open bracket

V = 2\pi \int\limits^4_0 {8x - x.x^{\frac{3}{2}}} \, dx

V = 2\pi \int\limits^4_0 {8x - x^{\frac{2+3}{2}}} \, dx

V = 2\pi \int\limits^4_0 {8x - x^{\frac{5}{2}}} \, dx

Integrate

V = 2\pi  * [{\frac{8x^2}{2} - \frac{x^{1+\frac{5}{2}}}{1+\frac{5}{2}}]\vert^4_0

V = 2\pi  * [{4x^2 - \frac{x^{\frac{2+5}{2}}}{\frac{2+5}{2}}]\vert^4_0

V = 2\pi  * [{4x^2 - \frac{x^{\frac{7}{2}}}{\frac{7}{2}}]\vert^4_0

V = 2\pi  * [{4x^2 - \frac{2}{7}x^{\frac{7}{2}}]\vert^4_0

Substitute 4 and 0 for x

V = 2\pi  * ([{4*4^2 - \frac{2}{7}*4^{\frac{7}{2}}] - [{4*0^2 - \frac{2}{7}*0^{\frac{7}{2}}])

V = 2\pi  * ([{4*4^2 - \frac{2}{7}*4^{\frac{7}{2}}] - [0])

V = 2\pi  * [{4*4^2 - \frac{2}{7}*4^{\frac{7}{2}}]

V = 2\pi  * [{64 - \frac{2}{7}*2^2^{*\frac{7}{2}}]

V = 2\pi  * [{64 - \frac{2}{7}*2^7]

V = 2\pi  * [{64 - \frac{2}{7}*128]

V = 2\pi  * [{64 - \frac{2*128}{7}]

V = 2\pi  * [{64 - \frac{256}{7}]

Take LCM

V = 2\pi  * [\frac{64*7-256}{7}]

V = 2\pi  * [\frac{448-256}{7}]

V = 2\pi  * [\frac{192}{7}]

V = [\frac{2\pi  * 192}{7}]

V = \frac{\pi  * 384}{7}

V = \frac{384}{7}\pi

Hence, the required volume is:

Volume = \frac{384}{7}\pi

You might be interested in
What is the image point of (9,−2) after a translation right 3 units and down 4 units?
-BARSIC- [3]

Answer:

(12,-6)

Step-by-step explanation:

I hope this helps and have a good day.

3 0
3 years ago
What’s 4/5 of 3 And what’s 2/9 of 20
Karo-lina-s [1.5K]

Answer:

4/5 of 3 = 12/5

2/9 of 20 = 40/9

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Can someone help me with number 3, please
oksian1 [2.3K]

Answer:

E

Step-by-step explanation:

We can see that section E of the graph increases, but unlike section B it is much longer and, therefore, increases more slowly.

Hope this helps friend.

4 0
2 years ago
Read 2 more answers
Plsss helppp brainkeistt reward
Mrac [35]

Answer:

1. D

2. B

3. A

4. B

6 0
3 years ago
PLEASE HURRY I'M BEING TIMED
kobusy [5.1K]

Answer:

i think the answer is D my guy

Step-by-step explanation:

do you need the steps??

mark brainliest :p

6 0
3 years ago
Other questions:
  • Andy sold 312 inGirl Scout cookies. If each box is 4 how many boxes did Andy sail
    15·1 answer
  • Maggie answered 20 out of 25 questions correctly on a math test. What percent of the questions did she answer correctly?​
    13·2 answers
  • The most important math skill for making change is
    13·1 answer
  • Select the relations that are functions.
    7·1 answer
  • Evaluate √81/√9<br> help lolll
    5·2 answers
  • Combine to create an equivalent expression -n + (-4) - (-4n) +6
    13·1 answer
  • A boat travels due north for 20km and then turns and travels due east for 7km. How far away is the boat from where it started?
    15·1 answer
  • Putting soil is sold in bags that contain 1/2 cubic foot of soil. Estimate how many bags Rylee should buy.
    11·2 answers
  • Timmy and Sarah head out to dinner and Garfinkels in Vaile. Timmy brings $40 with him. If buy two French Dips costing $10.95 a p
    15·1 answer
  • Please answer this i need helppp
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!