Find the Volume of the solid generated by revolving the region in the first quadrant

[Frenchi33]

y=sqrt(x-4), x=8 and y=0 about the y-axis:

I need to graph this in 3D on paper, but I'm confused... I know when I'm flipping the functions about the y-axis that I need to solve for x... So "y=sqrt(x-4)" would be "x=(y^2)+4", right? But do I graph the "x=(y^2)+4" or is that used only when I'm integrating with dy to calculate the Volume?

Or maybe I'm wrong all together?

 

[mmm4444bot]

y=sqrt(x-4), x=8 and y=0 about the y-axis:

I need to graph this in 3D on paper, but I'm confused... I know when I'm flipping the functions about the y-axis that I need to solve for x...

When you say "flipping the functions", I hope you are thinking "rotating the region". No function graph is being flipped (reflected), in this exercise.

If you're going to integrate along the y-axis (using washers, with dy), then yes, you need to express x in terms of y -- because x is the inner radius of the washers.

If you integrate along the x-axis (using shells, with dx), then use sqrt(x-4) for y -- because y is the height of the shells.

So "y=sqrt(x-4)" would be "x=(y^2)+4", right?

That's correct, for integrating with respect to dy.

But do I graph the "x=(y^2)+4" or is that used only when I'm integrating with dy to calculate the Volume

The graph of x=y^2+4, for y≥0, will look exactly the same as the graph of y=sqrt(x-4), for x≥4, because each of those functions describe the same relationship between x and y. (Don't confuse this with the concept of inverse functions; that's something else.)

The first expresses x as a function of y, and the second expresses y as a function of x, but the (x,y) pairs are the same for each. :cool:

 

[Frenchi33]

When you say "flipping the functions", I hope you are thinking "rotating the region". No function graph is being flipped (reflected), in this exercise.

If you're going to integrate along the y-axis (using washers, with dy), then yes, you need to express x in terms of y -- because x is the inner radius of the washers.

If you integrate along the x-axis (using shells, with dx), then use sqrt(x-4) for y -- because y is the height of the shells.


That's correct, for integrating with respect to dy.


The graph of x=y^2+4, for y≥0, will look exactly the same as the graph of y=sqrt(x-4), for x≥4, because both of those functions describe the same relationship between x and y. (Don't confuse this with the concept of inverse functions; that's something else.)

The first expresses x as a function of y, and the second expresses y as a function of x, but the (x,y) pairs are the same for each. :cool:

Oh, I see! Thanks for your help!

 

[mmm4444bot]

Were you able to graph the given information? Did you figure out the dimensions of the representative rectangle (slice to be rotated)? Which method did you decide to use?