Area, Arc Length, Volume & Curved Area of y=√x

[lockedup]

Homework Statement

For the curve y=\sqrt{x} , between x = 0and x = 2, find (a) the area under the curve, (b) the arc length, (c) the volume of the solid generated when the area is revolved about the x axis, (d) the curved area of this solid.

Homework Equations

ds = \sqrt {1+(y')^{2}}dx

The Attempt at a Solution

I did a and c pretty easily. My problem is arc length. I wrote out the integral \int^{2}_{0} \sqrt{1+\frac{1}{4x}}dx and realized I don't have a clue how to calculate it.

Also, for the "curved" area. Does my professor mean surface area? How do I do that?

 

[Dick]

To integrate that try a substitution like u^2=1+1/(4x). That will make it a rational function and then you can use partial fractions and stuff. If you don't 'know' how to solve a given integral, that's usually your first move. And the 'curved surface area' is area swept out by the curve. Look up 'surface of revolution'. There's a very similar formula to the one you have for arc length.