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lLovePhysics

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I have a question on the formulas for arc length and surface area.

Do you use the formula: [tex]s= \int_{c}^{d}\sqrt{1+[g'(y)]^2}dy[/tex] only when you are provided with a function x=g(y)?? Can you convert that to y=g(x) and solve it by replacing g'(y) with y(x), changing the bounds and the dy to dx?

Like for example if you wanted to solve:

Find the area of the surface formed by revolving the graph of [tex]f(x)=x^2[/tex] on the interval [0,2] about the y-axis.

How would you know whether to find the surface area with respect to x or y? Are these independent on the axis of revolution? Why does the book give the Surface area formula with respect to x and y?

Do you use the formula: [tex]s= \int_{c}^{d}\sqrt{1+[g'(y)]^2}dy[/tex] only when you are provided with a function x=g(y)?? Can you convert that to y=g(x) and solve it by replacing g'(y) with y(x), changing the bounds and the dy to dx?

Like for example if you wanted to solve:

Find the area of the surface formed by revolving the graph of [tex]f(x)=x^2[/tex] on the interval [0,2] about the y-axis.

How would you know whether to find the surface area with respect to x or y? Are these independent on the axis of revolution? Why does the book give the Surface area formula with respect to x and y?

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