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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

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What is the value of a in the equation 5a -10b = 45 when b = 3
leva [86]
Salutations!

5a-10b = 45

value of b = 3

Multiply 10 and 3

10*3=30

Now, you can solve the equation --

5a-30=45

separate the variables aside, and the numbers aside.

5a=45 +30

5a=75

a=75/5

a=15

Hope I helped (:

Have a great day!
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3 years ago
Please answer the question below (the question is question 64)
alexandr402 [8]
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A car traveld 32 miles in 4 hours at this rate how many miles will the car travel in 0.5 hours
Mandarinka [93]

Answer:

4 miles

Step-by-step explanation:

In order to find how many miles the car will travel in 0.5 hours, we can set up a proportion.

\frac{32}{4} = \frac{x}{0.5}

We can now cross multiply to find the value of x.

32 \cdot 0.5 = 16\\\\16\div4 = 4

So the car can travel 4 miles in 0.5 hours.

Hope this helped!

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3 years ago
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Which points are on the graph of the equation - 3x + 6y + 5 = -7?
ludmilkaskok [199]

Answer:

C and E

Step-by-step explanation:

To solve this problem, let's plug in each coordinates into the equation.

A:

  • -3(-3) + 6(6) + 5 = -7
  • 9 + 36 + 5 = -7
  • 50 = -7
  • 50 \neq -7
  • A. cannot be the answer, as 50 does not equal -7.

B:

  • -3(-2) + 6(0)+5=-7
  • 6+0+5=-7
  • 11=-7
  • 11\neq -7
  • B. cannot be the answer, as 11 does not equal -7.

C:

  • -3(0) + 6(-2) + 5 = -7
  • 0 - 12 + 5 = -7
  • -12 + 5 = -7
  • -7 = -7
  • C. can be the answer, as -7 does equal -7.

D:

  • -3(6)+6(-3)+5=-7
  • -18-18+5=-7
  • -36+5=-7
  • -31\neq -7
  • D. cannot be the answer, as -31 does not equal -7.

E:

  • -3(8) + 6(2) + 5 = -7
  • -24 + 12 + 5 = -7
  • -12 + 5 = -7
  • -7 = -7
  • E. can be the answer, as -7 does equal -7.
4 0
2 years ago
We may conclude that (a) events A and B are independent. (b) events A and B are mutually exclusive. (c) either A or B always occ
Yakvenalex [24]

Answer:

The answer is below

Step-by-step explanation:

The question is not complete. But I would show you how to solve the problem.

Two events A and B are said to be independent if the occurrence of event A does not affect the occurrence of event B and vice versa. P(A and B) = P(A) * P(B).

Two events A and B are said to be mutually exclusive if event A and event B cannot occur at the same time. P(A and B) = 0.

Two events A and B are said to be complementary when event A occurs if and only if event B does not occur and vice versa.

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2 years ago
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