Question

Sketch the region enclosed by the lines x=0, x=6, y=2, and y=6. Find the volume of the solid formed when the region is revolved around the y axis

Answer 1

By "revolved around the y axis", I assume it means y=0.

Assuming that, what do you get when you revolve a point at x=6 around the center at y=0?

Answer 2

y=6?

Answer 3

@qweqwe123123123123111, \(\text{y-axis}~\iff~x=0.\)

Answer 4

Thanks for the medal, but we've only just begun, and your first answer isn't quite right... :-)

When you revolve a point around another point, it encloses a plane figure. For instance, when the earth revolves around the sun, the earth encloses (or "draws", if you will) an area in the shape of an ellipse.

Answer 5

Y axis is vertical, X axis is horizontal

The X points are 0 and 6, so it's the x=6 point that gets revolved around the vertical Y axis

Answer 6

so when you revolve all the way around the y axis you will end up with a sphere?

Answer 7

or circle

Answer 8

No, we're talking about a plane here, so revolving a point around a center gives you a circle

Answer 9

ya

Answer 10

Yes!! Circle!! :-)

And what is the radius of that circle?

Answer 11

6

Answer 12

Right again!!! :-)

And do you know the formula for the area of a circle?

(and my next question will be "do you know the formula for the volume of a cylinder?")

Answer 13

The area of a circle is pi(r)^2
A cylinder is Bh or pi(r)^2*h

Answer 14

Excellent!! We're almost there!

So what's the area of the circle we just drew above?

Answer 15

113.1 or 36pi

Answer 16

Accurate to 1 decimal place, correct!! :-)

So all we have to do is add a "height" to that circle, and we have a cylinder.

The top and bottom of the cylinder are defined by the lines we drew intersecting the y axis.

So let's place our 113 inch circle at the bottom point y=2.

Where do we place the top of the cylinder?

Answer 17

y=6?

Answer 18

Yup!! You got it!!

Now here's the critical part: What is the distance between the points y=2 and y=6 ?

Answer 19

4

Answer 20

Which just also happens to be the height of our cylinder! :-)

So now we have the cylinder's height and we have the area of its base.

What's its volume?

Answer 21

452.4

Answer 22

Voila!!

You just figured out the answer to your problem!! :-)

Answer 23

Thank you so much!!! You have no idea how much you helped me!

Answer 24

No problem at all!! I'm glad I was able to help! :-)

Answer 25

I really appreciate it!!!