Volume of solid under graph and above circular region

Click For Summary
SUMMARY

The volume of the solid under the graph of z=sqrt(16-x^2-y^2) and above the circular region defined by x^2+y^2<=4 can be calculated using integration techniques. The correct approach involves integrating with respect to z rather than converting to polar coordinates. The integration should be set up as a double integral over the circular region, with the limits for r ranging from 0 to 2 and theta from 0 to 2π. This method will yield the accurate volume of the solid.

PREREQUISITES
  • Understanding of double integrals in calculus
  • Familiarity with polar coordinates
  • Knowledge of the equation of a sphere and its volume
  • Experience with integration techniques
NEXT STEPS
  • Study the method of integrating in cylindrical coordinates
  • Learn about volume calculations using double integrals
  • Explore the concept of horizontal slices in volume integration
  • Review examples of integrating functions over circular regions
USEFUL FOR

Students and educators in calculus, mathematicians focusing on volume calculations, and anyone interested in advanced integration techniques.

DSnead
Messages
1
Reaction score
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.
 
Physics news on Phys.org
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:
 

Similar threads

The volume of a "spherical cap" using triple integrals
  • · Replies 9 ·
Replies
9
Views
3K
Calculating the volume of the solid in this graph
  • · Replies 4 ·
Replies
4
Views
2K
Double Integral of function in region bounded by two circles
  • · Replies 19 ·
Replies
19
Views
3K
Triple Integral To Find Volume Between Cylinder And Sphere
  • · Replies 2 ·
Replies
2
Views
2K
[MultiVarCalc] Find the volume of the solid
  • · Replies 3 ·
Replies
3
Views
2K
Volume of solid region double integral
  • · Replies 27 ·
Replies
27
Views
3K
Change of variables in multiple integrals
  • · Replies 34 ·
2
Replies
34
Views
3K
How to find the volume when solids intersect?
  • · Replies 8 ·
Replies
8
Views
5K
Volume of revolution, region bounded by two functions
  • · Replies 1 ·
Replies
1
Views
2K
Volume between a region (above x-axis and below parabola) and a surface
  • · Replies 4 ·
Replies
4
Views
3K