Question

calc help

Answer 1

@ganeshie8

Answer 2

Have you tried sketching the three boundary curves as well as the axis of rotation? I'd strongly suggest you do so if you haven't already. Then, decide which of the three common methods for finding volumes of revolution would be best suited for application to this problem.

Hint: what are the coordinates of the y-intercept of y=e^(2x)?

Answer 3

its 0?

Answer 4

That's the x-coord. What's the y-coord.?

Answer 5

0,0

Answer 6

Think of what the graph of y=e^x looks like, and then re-examine your answer.

Answer 7

0,1

Answer 8

Enclose that within parentheses and you'll be all set.
Graph (0,1).
Graph the horiz. line y=1.
Graph the vert. line x=1.
Using a dashed line, indicate the axis of rotation.
Determine which method of finding the volume of rotation would be most suitable here.

Answer 9

washer method?

Answer 10

That would work. Disk meth. is out because this solid of rev. is hollow. That leaves one other possible method: which one?

Answer 11

Note that all of the answer choices end in "dx." That hints broadly at which method we must use.

Answer 12

its vertical

Answer 13

It = ?

Answer 14

That "dx" rules out use of the shell method (the shells would be horizontal and have thickness dy.

so, you'll need to write an expression in terms of x for the outer washer diameter. Fortunately, the inner washer diameter is a constant and is 2 (a vertical distance, measured from the axis of rot. y=-1 to the defining boundary y=1.

Have you drawn a sketch yet?

Answer 15

i graphed it on desmos

Answer 16

And what do you see?

Answer 17

Looks good. the region to be rotated about y=-1, to generate a solid of revolution, looks sort of triangular, with the lower left "vertex" located at (0,1). Agree or disagree?

Answer 18

agree

Answer 19

the outside radius is to be measured upward from the horiz. line y=-2. It will pass right thru the horiz line y=1 and end only when it "hits" the curve, that is, the graph of y=e^(2x).

Agree or disagree?

Answer 20

y=-1, not -2 right?

Answer 21

Yes, you're right. measure the outer radius from y=-1. Sorry about that.

Answer 22

th limit is from 0 to 1?

Answer 23

I'd phrase that as "we need to integrate from x=0 to x=1."

Answer 24

How do you find the area of a circle of radius R?

Answer 25

pi(r)^2

Answer 26

all right. You have millions of thin circles of radius R=1+e^(2x). Why the ' 1 ' in there?
What is the radius of the smaller (inner) circle (a hole runs thru the center of your washer)?

Answer 27

from y=-1 to y=1, so 2?

Answer 28

radius of the smaller circle is indeed 2, for the reason you've given.

So, how would you go about finding the area of a typical washer of outside radius R=1+e^(2x) and inside radius 2?

Answer 29

(R)^2-(r)^2, outer minus inner radius, both squared?

Answer 30

Yes. You'd take your expression, enclose the whole thing within par;entheses, and then multiply the result by pi. right?

Answer 31

so its b?

Answer 32

I'm reluctant to agree with "it's a, it's b, " answers. Could you possibly explain your choice?

Answer 33

the outer radius is (e^2x+1) and the inner radius us 2. and if you square both then subtract r from R, you get integral from 0 to 1 of ((e^2x+1)^2-2^2)dx then muliply it by pi, which you put before the integral

Answer 34

sounds as tho you have reasoned this problem out, which I greatly appreciate. If you've come this far, then I trust you to choose the correct answer from the four alternatives. Are y ou all clear yourself on what we have discussed?

Answer 35

yes ty

Answer 36

My pleasure. Take care, hope to work with you again! 'Night.