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Volume of a bowl (using variables not numbers)

  • Thread starter htoor9
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  • #1
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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
 

Answers and Replies

  • #2
33,496
5,188

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.
 
  • #3
17
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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
 
  • #4
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5,188
The bottom point of the bowl is at (0, -a).
 
  • #5
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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?
 
  • #6
33,496
5,188
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?
 

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