Question

im stumped calculus circumference arc. how come we cant use integral C(x)dx to find surfacea area. C(x) is the circumference so its 2pi*r , and dx is the thickness.
so why isnt surface area integral 2pi * r * dx
instead it is integral 2pi y sqrt ( 1 + dy/dx^2)dx

Answer 1

You're trying to compute the area of a circle?

Answer 2

so say you have f(x)>0 in the first quadrant

Answer 3

im doing the formula for surface area , made by revolving a curve about an axis .

Answer 4

generated i should say

Answer 5

Ok

Answer 6

I don't understand that second integral you wrote.

Answer 7

Oh wait, yes I do.

Answer 8

integral 2pi * f(x) sqrt ( 1 + f ' (x) ^2 ) dx

Answer 9

yeah

Answer 10

Polpak, when you get finished can you assist me?

Answer 11

so my question is , why cant we just use the easier formula
integral 2pi f(x) dx

Answer 12

what r u guys parternes?

Answer 13

Are you rotating around the x axis?

Answer 14

yes in this case

Answer 15

say y = x^2 from 0 to 2 , but trying to find surfacea area

Answer 16

But you still need to account for the way f(x) is changing over your dx

Answer 17

You would be fine if it were flat washers

Answer 18

But along the top edge there can be a lot of slopes that will change the area of the final surface.

Answer 19

thats true but

Answer 20

when you go to zero, dont they approach 2pi * r , the height

Answer 21

Yes, but they still have an angle.

Answer 22

You're ok computing the volume that way

Answer 23

but not the surface area.

Answer 24

actually its not volume

Answer 25

it would be the surface of a cylinder type figure, i guess, not sure

Answer 26

for example here, we have using my way integral 2pi x^2 dx from 0 to 2

Answer 27

See you're way makes the assumption that along the top edge for a tiny dx, that the length of the line of the function = dx.

Answer 28

the books way is integral 2pi x^2 sqrt ( 1 + (2x)^2) dx

Answer 29

come back

Answer 30

but it doesn't. The length of the line for a small dx is proportional the the square root of the square of dy^2 + dx^2

Answer 31

err I said that wrong

Answer 32

But you see what I mean.

Answer 33

We are ok with the area under the curve assuming that as we take smaller and smaller values for dx, the area approaches f(x) * dx

Answer 34

But for the line integral it doesn't come close

Answer 35

yeah, why is that

Answer 36

Let me see if I can explain.

Answer 37

http://www.dabbleboard.com/draw?b=Guest666913&i=0&c=d86fb2255acc9e8490af7e397d12fe9fc183a40a

Answer 38

You there?

Answer 39

http://www.twiddla.com/529576