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
BARSIC [14]
3 years ago
13

How do you find the volume of the solid generated by revolving the region bounded by the graphs

Mathematics
1 answer:
d1i1m1o1n [39]3 years ago
6 0

Answer:

About the x axis

V = 4\pi[ \frac{x^5}{5}] \Big|_0^2 =4\pi *\frac{32}{5}= \frac{128 \pi}{5}

About the y axis

V = \pi [4y -y^2 +\frac{y^3}{12}] \Big|_0^8 =\pi *\frac{32}{3}= \frac{32 \pi}{3}

About the line y=8

V = \pi [64x -\frac{32}{3}x^3 +\frac{4}{5}x^5] \Big|_0^2 =\pi *(128-\frac{256}{3} +\frac{128}{5})= \frac{1024 \pi}{5}

About the line x=2

V = \frac{\pi}{2} [\frac{y^2}{2}] \Big|_0^8 =\frac{\pi}{4} *(64)= 16\pi

Step-by-step explanation:

For this case we have the following functions:

y = 2x^2 , y=0, X=2

About the x axis

Our zone of interest is on the figure attached, we see that the limit son x are from 0 to 2 and on  y from 0 to 8.

We can find the area like this:

A = \pi r^2 = \pi (2x^2)^2 = 4 \pi x^4

And we can find the volume with this formula:

V = \int_{a}^b A(x) dx

V= 4\pi \int_{0}^2 x^4 dx

V = 4\pi [\frac{x^5}{5}] \Big|_0^2 =4\pi *\frac{32}{5}= \frac{128 \pi}{5}

About the y axis

For this case we need to find the function in terms of x like this:

x^2 = \frac{y}{2}

x = \pm \sqrt{\frac{y}{2}} but on this case we are just interested on the + part x=\sqrt{\frac{y}{2}} as we can see on the second figure attached.

We can find the area like this:

A = \pi r^2 = \pi (2-\sqrt{\frac{y}{2}})^2 = \pi (4 -2y +\frac{y^2}{4})

And we can find the volume with this formula:

V = \int_{a}^b A(y) dy

V= \pi \int_{0}^8 2-2y +\frac{y^2}{4} dy

V = \pi [4y -y^2 +\frac{y^3}{12}] \Big|_0^8 =\pi *\frac{32}{3}= \frac{32 \pi}{3}

About the line y=8

The figure 3 attached show the radius. We can find the area like this:

A = \pi r^2 = \pi (8-2x^2)^2 = \pi (64 -32x^2 +4x^4)

And we can find the volume with this formula:

V = \int_{a}^b A(x) dx

V= \pi \int_{0}^2 64-32x^2 +4x^4 dx

V = \pi [64x -\frac{32}{3}x^3 +\frac{4}{5}x^5] \Big|_0^2 =\pi *(128-\frac{256}{3} +\frac{128}{5})= \frac{1024 \pi}{5}

About the line x=2

The figure 4 attached show the radius. We can find the area like this:

A = \pi r^2 = \pi (\sqrt{\frac{y}{2}})^2 = \pi\frac{y}{2}

And we can find the volume with this formula:

V = \int_{a}^b A(y) dy

V= \frac{\pi}{2} \int_{0}^8 y dy

V = \frac{\pi}{2} [\frac{y^2}{2}] \Big|_0^8 =\frac{\pi}{4} *(64)= 16\pi

You might be interested in
A customer opens a checking account and a savings account at a bank. They will deposit a maximum of $600, some in the checking a
bagirrra123 [75]

Answer:

x +y \leq \\ 600

Step-by-step explanation:

8 0
2 years ago
Write the point- slope form of the equation of the line through the given point with the given slope
pashok25 [27]

Answer:

y=mx+b

y=-1x-4

y=2x+4

Step-by-step explanation:

8 0
3 years ago
No bots answer fast plz 1. (4x + 8) + (7x + 3) =
Delicious77 [7]
12x + 11

………………………………….
3 0
2 years ago
Read 2 more answers
Thank you for your help
olya-2409 [2.1K]

Answer:

I'm not completely sure but if they want exponents for 6 * 6 * 6 then it should be 6^{3}

Step-by-step explanation:

5 0
2 years ago
I NEED HELP: So this question came on a quiz I'm taking and the answer I got was $1,723.45 but it says its incorrect can someone
Olin [163]

Answer: the answer is right, try putting a $ in front of it, sometimes those programs mess up if you aren't specific

5 0
2 years ago
Other questions:
  • On a piece of paper, graph f(x) =
    7·1 answer
  • Which choice answer is it?
    9·1 answer
  • Please helpppppppppp me
    13·1 answer
  • What is k in terms of n?
    5·2 answers
  • Help me ASAP.<br><br> What function is graphed below?
    5·1 answer
  • . A company is designing a new cylinder. The volume of the cylinder can be represented by the following equation, y = z · π(10 +
    14·1 answer
  • Help please lol thankssss
    6·1 answer
  • (I BEG U TUTORS HALP) the linear equation. Y= -1/3 x+5 is shown graphed. If it was combined with the equation y=5/3 x-3 to form
    9·1 answer
  • Help please i’m very confused!
    9·2 answers
  • PLEASE HELP ME ILL MAKE BRAINLIEST!!!!! <br> I need the correct answer
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!