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
PolarNik [594]
1 year ago
6

Use polar coordinates to find the volume of the given solid. Inside both the cylinder x2 y2 = 1 and the ellipsoid 4x2 4y2 z2 = 6

4
Mathematics
1 answer:
Anton [14]1 year ago
3 0

The Volume of the given solid using polar coordinate is:\frac{-1}{6} \int\limits^{2\pi}_ {0} [(60) ^{3/2} \; -(64) ^{3/2} ] d\theta

V= \frac{-1}{6} \int\limits^{2\pi}_ {0} [(60) ^{3/2} \; -(64) ^{3/2} ] d\theta

<h3>What is Volume of Solid in polar coordinates?</h3>

To find the volume in polar coordinates bounded above by a surface z=f(r,θ) over a region on the xy-plane, use a double integral in polar coordinates.

Consider the cylinder,x^{2}+y^{2} =1 and the ellipsoid, 4x^{2}+ 4y^{2} + z^{2} =64

In polar coordinates, we know that

x^{2}+y^{2} =r^{2}

So, the ellipsoid gives

4{(x^{2}+ y^{2)} + z^{2} =64

4(r^{2}) + z^{2} = 64

z^{2} = 64- 4(r^{2})

z=± \sqrt{64-4r^{2} }

So, the volume of the solid is given by:

V= \int\limits^{2\pi}_ 0 \int\limits^1_0{} \, [\sqrt{64-4r^{2} }- (-\sqrt{64-4r^{2} })] r dr d\theta

= 2\int\limits^{2\pi}_ 0 \int\limits^1_0 \, r\sqrt{64-4r^{2} } r dr d\theta

To solve the integral take, 64-4r^{2} = t

dt= -8rdr

rdr = \frac{-1}{8} dt

So, the integral  \int\ r\sqrt{64-4r^{2} } rdr become

=\int\ \sqrt{t } \frac{-1}{8} dt

= \frac{-1}{12} t^{3/2}

=\frac{-1}{12} (64-4r^{2}) ^{3/2}

so on applying the limit, the volume becomes

V= 2\int\limits^{2\pi}_ {0} \int\limits^1_0{} \, \frac{-1}{12} (64-4r^{2}) ^{3/2} d\theta

=\frac{-1}{6} \int\limits^{2\pi}_ {0} [(64-4(1)^{2}) ^{3/2} \; -(64-4(2)^{0}) ^{3/2} ] d\theta

V = \frac{-1}{6} \int\limits^{2\pi}_ {0} [(60) ^{3/2} \; -(64) ^{3/2} ] d\theta

Since, further the integral isn't having any term of \theta.

we will end here.

The Volume of the given solid using polar coordinate is:\frac{-1}{6} \int\limits^{2\pi}_ {0} [(60) ^{3/2} \; -(64) ^{3/2} ] d\theta

Learn more about Volume in polar coordinate here:

brainly.com/question/25172004

#SPJ4

You might be interested in
Which is the better buy? Show your work and explain your reasoning.
Vinvika [58]

Answer:

5.03

Step-by-step explanation:

Because

6 0
2 years ago
1. Add: 85 + (-96) *<br> 2. Subtract: -7 – 4 *<br> 3. Multiply: 7(-12) *<br> 4. Divide: 6÷ (-18) *
Monica [59]

Answer:

1 ._11

2.11

3._84

4._0.333

6 0
2 years ago
Read 2 more answers
Calculate axis of symmetry and vertex for equation 4x squared + 8x = 32
Sloan [31]

Answer:

x=2,−4 hope this was help full

6 0
3 years ago
HELP PLEASE!! The following data shows the weight in ounces of 10 different bags of candy.
STatiana [176]

Answer:

C.

Mean= 4.9

Mean Absolute Deviation (MAD): 4.099173553719

Step-by-step explanation:

An outlier is a value that is very different from the other data in your data set. This can skew your results. As you can see, having outliers often has a significant effect on your mean and standard deviation. Because of this, we must take steps to remove outliers from our data sets.

outlier: 21

8 0
2 years ago
Pregunta 14
IrinaK [193]

Answer:

sorry for points

Step-by-step explanation:

6 0
2 years ago
Other questions:
  • I NEED HELP PLEASE FAST ILL PICK BRAINLIEST AND A LOT OF POINTS PLZ
    14·2 answers
  • If UT is a midsegment of QRS, find SQ<br> A. 8 <br> B. 10 <br> C. 16 <br> D. 20
    5·1 answer
  • What is the y-intercept of the linear equation y=5x+7
    10·2 answers
  • You make 24 cookies using 2 cups of sugar
    14·2 answers
  • How can you use 260 divided by 13 to help you find 286 divided by 13
    6·1 answer
  • Help please thanks so much
    9·1 answer
  • For each interval , decide if the average rate change is positive or negative
    13·1 answer
  • It about geometry and math that’s it
    14·1 answer
  • Describe the zeros of the graphed function.
    6·1 answer
  • X+2y=5 and 4x-12y=-20 solve using elimination and substitution
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!