# Volume of a bowl (using variables not numbers)

## 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

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Mark44
Mentor

## 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.

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

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
Mentor
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?