Question

Find the area of the surface by rotating the graph of f(x) = sqrt(6-x) about x-axis from x = 0 to x = 4

Answer 1

@mtbender74

Answer 2

http://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx

Answer 3

How would we start? I can't graph anything >.<

Answer 4

ok...i'm going to build off of the link from bob

Answer 5

did you look at it?

Answer 6

i did

Answer 7

so we need to first build the pieces we need...

Answer 8

ok let's begin

Answer 9

\[S=\int2\pi(\sqrt{6-x})ds\]
so we need to calculate ds first

Answer 10

\[ds=\sqrt{1+(\frac{dy}{dx})^{2}}dx\]

Answer 11

so dy/dx is the derivative of your function...let's see if we can get ds...

Answer 12

what have you gotten for ds?

Answer 13

one second

Answer 14

I got a double integral. I don't think I did something right.

Answer 15

What are you getting?

Answer 16

I've been having problems with the site today...sorry.

Answer 17

were you able to calculate ds?

Answer 18

I ended up with a double integral, which I doubt is the correct answer

Answer 19

no...i got a single integral...let me walkthrough my steps and see if you can go from there

Answer 20

ok

Answer 21

\[y=(6-x)^{1/2}\]
\[\frac{dy}{dx}=\frac{-1}{2(6-x)^{1/2}}\]
\[ds=\sqrt{1+(\frac{dy}{dx})^{2}}dx=\sqrt{1+(\frac{-1}{2(6-x)^{1/2}})^{2}}dx\]

Answer 22

\[ds=\sqrt{1+\frac{1}{4(6-x)}}dx=\sqrt{1+\frac{1}{24-4x}}dx\]
\[=\sqrt{\frac{24-4x+1}{24-4x}}dx=\sqrt{\frac{25-4x}{24-4x}}dx\]
\[=\frac{\sqrt{25-4x}}{2\sqrt{6-x}}dx\]

Answer 23

all of that make sense?

Answer 24

Yeah, so far.
should i solve for the one at the bottom?

Answer 25

no...that's just ds...now we put that into the surface area formula, simplify, and integrate...

Answer 26

\[S=2\pi\int_{0}^{4}yds=2\pi\int_{0}^{4}(6-x)^{1/2}(\frac{\sqrt{25-4x}}{2(6-x)^{1/2}})dx\]

Answer 27

the (6-x)^(1/2) terms cancel, the 2 in the denominator cancels the 2 on the outside...

Answer 28

leaving you with
\[S=\pi\int_{0}^{4}(25-4x)^{1/2}dx\]

Answer 29

can you integrate that?

Answer 30

I think so. one second

Answer 31

49pi/3

Answer 32

hmmm...not what i got can you show your steps?

Answer 33

when you do the integration, you should get \[-\frac{1}{6}\pi(25-4x)^{3/2}\]which gets evaluated from 0 to 4