Finding out the amount of glass through integration

[orangesun]

Homework Statement

A glass vase has the shape of the solid obtained by rotating about the y–axis the area in
the first quadrant lying over the x–interval [0,a] and under the graph of y = $$\sqrt{x}$$
Determine how much glass is contained in the vase.

Homework Equations

y = $$\sqrt{x}$$

The Attempt at a Solution

I know the integration to find the area under the graph would be
F = 2/3x^(3/2)

Area = $$\int2/3x^(3/2)$$ from 0 to a
and you would need to times the total area by 2 to get the full "vase"


Thanks a lot!

 

[Mark44]

This is not a precalculus problem. Post calculus problems in the Calculus & Beyond section.

Homework Statement

A glass vase has the shape of the solid obtained by rotating about the y–axis the area in
the first quadrant lying over the x–interval [0,a] and under the graph of y = $$\sqrt{x}$$
Determine how much glass is contained in the vase.

Homework Equations

y = $$\sqrt{x}$$

The Attempt at a Solution

I know the integration to find the area under the graph would be
F = 2/3x^(3/2)

This is the antiderivative of x^1/2. To get the area, you need a definite integral.

However, this problem is not asking for the "area" of glass. It's asking for the volume of glass.

Area = $$\int2/3x^(3/2)$$ from 0 to a
and you would need to times the total area by 2 to get the full "vase"

This makes no physical sense. You are attempting to take the antiderivative of a function that is the antiderivative of x^1/2.

This problem involves calculating the volume of revolution. There are two ways to calculate a volume of revolutions: breaking the solid up into disks or breaking the solid up into thin cylindrical shells. Your textbook should have examples of each of these techniques.

Thanks a lot!