Evaluate 4x² + 3x for x = 2

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

Anton [14]

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:

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3 0

2 years ago

You go to the pet store and buy a goldfish which sells for $2.00. The store charges 9% sales tax. How much tax will you pay?

RideAnS [48]

Answer:

tax=$0.18

Step-by-step explanation:

total cost = $2.18

3 0

2 years ago

Find f(x) if it is known that f(x-2)=2x-4

Nikolay [14]

Answer:

f=2

Step-by-step explanation:

f(x-2)=2x-4

divide x-2 to 2x-4

f=2

f(2-2)=4-4

3 0

3 years ago

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally

mariarad [96]

Answer:

a) 0.1587 (or 15.87%)

b) 0.0478 (or 4.78%)

c) 0.7935 (or 79.35%)

Step-by-step explanation:

Find the following probabilities:

(a) the thickness is less than 3.0 mm

We need to find the area under the Normal curve with a mean of 4.5 mm and a standard deviation of 1.5 mm to the left of 3 mm

<h3>(See picture 1 attached) </h3>

We can do that with the help of a calculator or a spreadsheet.

<em>In Excel use </em>

<em>NORMDIST(3,4.5,1.5,1) </em>

<em>In OpenOffice Calc use </em>

<em>NORMDIST(3;4.5;1.5;1) </em>

and we get the value 0.1587 (or 15.87%)

(b) the thickness is more than 7.0 mm

Now we need the area to the right of 7 (1 - the area to the left of 7)

<h3>(See picture 2 attached) </h3>

<em>In Excel use </em>

<em>1-NORMDIST(7,4.5,1.5,1) </em>

<em>In OpenOffice Calc use </em>

<em>1-NORMDIST(7;4.5;1.5;1) </em>

and we get the value 0.0478 (or 4.78%)

(c) the thickness is between 3.0 mm and 7.0 mm

We are looking for the area between 3 and 7

<h3>(See Picture 3) </h3>

Since the area under the Normal equals 1, we have

Area to the left of 3 + Area between 3 and 7 + Area to the right of 7 = 1

Hence,

0.1587 + Area between 3 and 7 + 0.0478 = 1

and

Area between 3 and 7 = 1 - 0.1587 - 0.0478 = 0.7935 (or 79.35%)

4 0

3 years ago

Read 2 more answers

The double number line shows how far a hummingbird can fly in 1 second -

jarptica [38.1K]

Answer:

1

3

130

Step-by-step explanation:

as shown on the lines, it takes the bird one second to fly 13 meters. the answer to the first blank is 1

using this ratio, 13 meters per second, we can find how long it takes for the bird to fly 39 meters.

13/1=39/x

this is 13x=39

x=3 seconds

use the same ratio to find distance

13/1=x/10

13=x/10

130=x

3 0

3 years ago