Volume of solid under graph and above circular region

In summary, the conversation is about finding the volume of a solid under the graph of z=sqrt(16-x^2-y^2) and above a circular region in the xy plane. The method of using polar coordinates is brought up, but it is suggested to instead divide into horizontal slices and integrate over z.
  • #1
DSnead
1
0
Find the volume of the solid under the graph of z=sqrt(16-x^2-y^2) and above the circular region x^2+y^2<=4 in the xy plane

I know I must go to polar. So z=sqrt(16-r^2). Does r range from 0-2? I am not sure what theta ranges from (0-2pi)? I set up the integral as int(int r*sqrt(16-r^2), r=0..2), theta=0..2pi) but I get nowhere near any of the answer choices.
 
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  • #2
Welcome to PF!

Hi DSnead! Welcome to PF! :smile:
DSnead said:
… I know I must go to polar …

No, just divide into horizontal slices of thickness dz, and integrate over z. :wink:
 

1. What is the formula for finding the volume of a solid under a graph and above a circular region?

The formula for finding the volume of a solid under a graph and above a circular region is given by: V = π∫ab(f(x))^2dx, where a and b are the limits of the circular region and f(x) is the function representing the graph.

2. How do you determine the limits of integration for finding the volume of the solid?

The limits of integration for finding the volume of the solid depend on the given circular region. The lower limit (a) is the starting point of the circular region on the x-axis and the upper limit (b) is the ending point. These limits can be determined by analyzing the given graph and identifying the points where the circular region begins and ends.

3. Can the volume of the solid under a graph and above a circular region be negative?

No, the volume of a solid cannot be negative. The volume of the solid under a graph and above a circular region represents the amount of space that is enclosed by the graph and the circular region. Since space cannot have a negative value, the volume of the solid cannot be negative.

4. What units are used to measure the volume of the solid under a graph and above a circular region?

The volume of the solid is usually measured in cubic units, such as cubic centimeters (cm3) or cubic meters (m3). However, the units may vary depending on the given problem and the units used for the measurements of the circular region and the graph.

5. Can the volume of the solid under a graph and above a circular region be calculated using any shape?

No, the volume of the solid can only be calculated using a circular region and a graph that is a function of x. If the shape of the circular region or the graph is different, a different formula may need to be used to find the volume of the solid.

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