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IgorLugansk [536]

3 years ago

11

A point A is graphed at (0, 3). Complete the coordinates of a shape that can be rotated about the y-axis to create a cylinder. P

rovide an explanation for your answer to receive full credit.

1 answer:

Molodets [167]3 years ago

7 0

To rotate a shape around the Y axis to form a cylinder, you would need two points on the same X axis,

When the shape is rotated it needs to be a cylinder, so the shape you need has to be either a square or a rectangle.

you can pick any X and Y coordinates that would make a square or rectangle as long as one other X is the same as Point A.

You can use Point B (0, 0), Point C (1,0) and Point D (1,3)

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Can anyone help me integrate :

worty [1.4K]

Rewrite the second factor in the numerator as

$2x^2+6x+1=2(x+2)^2-2(x+2)-3$

Then in the entire integrand, set $x+2=\sqrt3\sec t$ , so that $\mathrm dx=\sqrt3\sec t\tan t\,\mathrm dt$ . The integral is then equivalent to

$\displaystyle\int\frac{(\sqrt3\sec t-2)(6\sec^2t-2\sqrt3\sec t-3)}{\sqrt{(\sqrt3\sec t)^2-3}}(\sqrt3\sec t)\,\mathrm dt$
$=\displaystyle\int\frac{(6\sqrt3\sec^3t-18\sec^2t+\sqrt3\sec t+6)\sec t}{\sqrt{\sec^2t-1}}\,\mathrm dt$
$=\displaystyle\int\frac{(6\sqrt3\sec^3t-18\sec^2t+\sqrt3\sec t+6)\sec t}{\sqrt{\tan^2t}}\,\mathrm dt$
$=\displaystyle\int\frac{(6\sqrt3\sec^3t-18\sec^2t+\sqrt3\sec t+6)\sec t}{|\tan t|}\,\mathrm dt$

Note that by letting $x+2=\sqrt3\sec t$ , we are enforcing an invertible substitution which would make it so that $t=\mathrm{arcsec}\dfrac{x+2}{\sqrt3}$ requires $0\le t$ or $\dfrac\pi2$ . However, $\tan t$ is positive over this first interval and negative over the second, so we can't ignore the absolute value.

So let's just assume the integral is being taken over a domain on which $\tan t>0$ so that $|\tan t|=\tan t$ . This allows us to write

$=\displaystyle\int\frac{(6\sqrt3\sec^3t-18\sec^2t+\sqrt3\sec t+6)\sec t}{\tan t}\,\mathrm dt$
$=\displaystyle\int(6\sqrt3\sec^3t-18\sec^2t+\sqrt3\sec t+6)\csc t\,\mathrm dt$

We can show pretty easily that

$\displaystyle\int\csc t\,\mathrm dt=-\ln|\csc t+\cot t|+C$
$\displaystyle\int\sec t\csc t\,\mathrm dt=-\ln|\csc2t+\cot2t|+C$
$\displaystyle\int\sec^2t\csc t\,\mathrm dt=\sec t-\ln|\csc t+\cot t|+C$
$\displaystyle\int\sec^3t\csc t\,\mathrm dt=\frac12\sec^2t+\ln|\tan t|+C$

which means the integral above becomes

$=3\sqrt3\sec^2t+6\sqrt3\ln|\tan t|-18\sec t+18\ln|\csc t+\cot t|-\sqrt3\ln|\csc2t+\cot2t|-6\ln|\csc t+\cot t|+C$
$=3\sqrt3\sec^2t-18\sec t+6\sqrt3\ln|\tan t|+12\ln|\csc t+\cot t|-\sqrt3\ln|\csc2t+\cot2t|+C$

Back-substituting to get this in terms of $x$ is a bit of a nightmare, but you'll find that, since $t=\mathrm{arcsec}\dfrac{x+2}{\sqrt3}$ , we get

$\sec t=\dfrac{x+2}{\sqrt3}$
$\sec^2t=\dfrac{(x+2)^2}3$
$\tan t=\sqrt{\dfrac{x^2+4x+1}3}$
$\cot t=\sqrt{\dfrac3{x^2+4x+1}}$
$\csc t=\dfrac{x+2}{\sqrt{x^2+4x+1}}$
$\csc2t=\dfrac{(x+2)^2}{2\sqrt3\sqrt{x^2+4x+1}}$

etc.

3 0

3 years ago

What is |–14|<br>A. 1/14<br>B. -7<br>C. -14 <br>D. 7<br>E. 14

expeople1 [14]

Answer:

14

Step-by-step explanation:

absolute value works where it doesnt matter if the number is positive or negative. the absolute value is the number positive. an absolute value can not be negative.

5 0

3 years ago

A geography teacher lists 12 South American countries on the chalkboard. She asks each student in her class to write a report co

exis [7]

There 220 ways to select 3 of the 12 South American countries for the Geography assignment that solved by using the theory of combination in mathematics. The combination in mathematics is a way of choosing items from a set of items. The combination formula is n x (n-1) x ... x (n- k + 1) / k x (k-1) x ... x 1 where n is the number items in the set which is 12 and k is the amount you want to combine from the set which is 3.

4 0

3 years ago

Please help on this really need help on this

ira [324]

reflection across the x axis and rotation

5 0

3 years ago

Read 2 more answers

10 points<br><br> Someone help with these problems?

Mama L [17]

Answer:

Step-by-step explanation:

Q1 Equation-----12n=384 where n is no. Of weeks

Solution ---- n=32

Q2 Equation-------36W=576

Solution --------W =16

4 0

3 years ago