Volume of a bowl (using variables not numbers)

[htoor9]

Homework Statement

A hemispherical bowl of radius "a" contains water to a depth "h"

Fine the volume of water in the bowl.

Homework Equations

Pi * Integral(R(x))^2

basically the disk method

The Attempt at a Solution

The answer is (Pi*h²*(3a-h))/3...i just have no idea how to set up the integral to get that. Thanks guys

 

[Mark44]

Homework Statement

A hemispherical bowl of radius "a" contains water to a depth "h"

Fine the volume of water in the bowl.

Homework Equations

Pi * Integral(R(x))^2

basically the disk method

The Attempt at a Solution

The answer is (Pi*h²*(3a-h))/3...i just have no idea how to set up the integral to get that. Thanks guys

Start by sketching a graph of the function that represents the bowl. I recommend sketching the lower half of the graph of x² + y² = a² to make your computations a little simpler.

Draw a line across the bowl representing a level of h units. Since h is measured from the bottom of the bowl, if you have located the bowl as I suggested, the water level crosses the y-axis at (0, h - a). For example, if the bowl's radius is 6'' and the water is 2" deep, the water level on the y-axis is at (0, 2 - 6) = (0, -4).

Draw a typical "slice" of the water. What is its volume? That's what you will use for your integrand.

 

[htoor9]

Start by sketching a graph of the function that represents the bowl. I recommend sketching the lower half of the graph of x² + y² = a² to make your computations a little simpler.

Draw a line across the bowl representing a level of h units. Since h is measured from the bottom of the bowl, if you have located the bowl as I suggested, the water level crosses the y-axis at (0, h - a). For example, if the bowl's radius is 6'' and the water is 2" deep, the water level on the y-axis is at (0, 2 - 6) = (0, -4).

Draw a typical "slice" of the water. What is its volume? That's what you will use for your integrand.

since I'm revolving it around the y axis, how do i know what the length of the axis to the end of the bowl with the water is? I'm really confused

 

[Mark44]

The bottom point of the bowl is at (0, -a).

 

[htoor9]

The bottom point of the bowl is at (0, -a).

ok. i still can't figure out what the equation of the curve is... is it h^2?

 

[Mark44]

An equation has = in it. Your graph should show the lower half of a circle whose center is at the origin, and whose radius is a. You know the equation of a circle, right?