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
Doss [256]
3 years ago
8

Use cylindrical coordinates. find the volume of the solid that is enclosed by the cone z = x2 + y2 and the sphere x2 + y2 + z2 =

72.
Mathematics
1 answer:
sashaice [31]3 years ago
7 0
Let R be the solid. Then the volume is

\displaystyle\iiint_R\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta

which follows from the facts that

\begin{cases}x=r\cos\theta\\y=r\sin\theta\\z=\zeta\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta
(by computing the Jacobian)

and

z=x^2+y^2=r^2\implies z+z^2=72\implies z=-9\text{ or }z=8
(we take the positive solution, since it's clear that R lies above the x-y plane)
r^2+z^2=72\implies z=\pm\sqrt{72-r^2}
(again, taking the positive root for the same reason)
z=r^2\implies 8=r^2\implies r=\sqrt8

\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta
=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr
=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}r(\sqrt{72-r^2}-r^2)\,\mathrm dr
=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}(r\sqrt{72-r^2}-r^3)\,\mathrm dr
=\dfrac{32(27\sqrt2-35)\pi}3
You might be interested in
Is mass proportional to the force of gravity??? <br> PLZZ HELP WILL MARK THE BRAINLIEST
garri49 [273]
Noooo definitely not.
5 0
3 years ago
Read 2 more answers
In the base 5 number system, which number comes after 444,444?
RSB [31]
I think it’s 555,555






3 0
3 years ago
How many times can 27 go into 192
lutik1710 [3]
It can go into 192 7.1 times
7 0
3 years ago
Read 2 more answers
Write a rule for the linear function in the table.
JulijaS [17]

Answer:

I guess that you want to know the transformations:

We start with:

f(x) = y = 4*x + 3

a)the transformed function is:

f(x) = y = -4*x - 3

So the sign changed.

This means that we go from (x, y) to (x, - y)

This is a reflection over the x-axis which changes the sin of the y component.

b) Now we go to f(x) = 4*x + 3

So the coefficient in the leading term changed.

This is a horizontal contraction:

A horizontal contraction of factor K for the function g(x) is: g(K*x)

In our case, we have:

f(K*x) = 4*(k*x) + 3 = x + 3

4*k*x = x

4*k = 1

k = 1/4

Then the transformation is an horizontal contraction of scale factor 1/4.

6 0
3 years ago
The charge for I’ve skating is $3 for the skate rental and $2 per hour to skate. How much will you pay if you skate for 3 1/2 ho
sp2606 [1]

Answer:

$11

Step-by-step explanation:

multiply $2 by four because they charge by hour and not by half hour. = 8

add $3 to 8. =11

4 0
3 years ago
Other questions:
  • Given the graph below, find MN. Round to the nearest hundredth.
    9·1 answer
  • HELLLLPPPP!!!!! NEED ANSWER NOW, Judging from past basketball games, the probability that you make a free throw is 4/5. If you g
    13·1 answer
  • Use the triangle below and the given trigonometric ratio to determine which angle in the triangle below is
    13·1 answer
  • Ezra's dad is building a cover for his sandbox. The sandbox is in the shape of a kite as shown. A kite has a width of 6 feet and
    11·1 answer
  • How many triangles can fit into one triangle
    10·2 answers
  • Consider two points on the PPF: point A, at which there are 50 oranges and 100 apricots, and point B, at which there are 51 oran
    15·1 answer
  • Can someone explain?​
    6·1 answer
  • What is the equation of the linear function represented by the table?
    10·1 answer
  • Please simplify this
    9·1 answer
  • 5 tons of fertilizer cost $6,600.00. What is the price per pound?
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!