Triple Integrals with Cylindrical Coordinates

In summary, the integral in question involves evaluating the function 2(x^3+xy^2) over the solid E, which lies beneath the paraboloid z = 9 - x2 - y2 in the first octant. Converting to cylindrical coordinates and correcting the bounds, the integral can be solved to get the correct answer.
  • #1

Homework Statement


Evaluate the integral, where E is the solid in the first octant that lies beneath the paraboloid z = 9 - x2 - y2.

∫∫∫(2(x^3+xy^2))dV

Homework Equations



x=rcosθ
y=rsinθ
x^2+y^2=r^2

The Attempt at a Solution



θ=0 to 2π, r=0 to 3, z=0 to (9-r^2)

2(x^3+xy^2)=2x(x^2+y^2)=2rcos(θ)(r^2)

∫0 to 2π ∫0 to 3 ∫0 to (9-r^2) (2rcos(θ)r^2)rdzdrdθ

I was wondering if my bounds were correct. And when I solved the integral I keep getting an answer of 0, which is incorrect. Can someone please help me with this problem?
 
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  • #2
If you're integrating over the first octant, then theta doesn't go from 0 to 2pi. Other than that, the rest is ok.
 
  • #3
Thank you so much! That was the problem. I missed the part where it said the first octant.
 

1. What are cylindrical coordinates and how are they different from Cartesian coordinates?

Cylindrical coordinates are a coordinate system used to describe points in a three-dimensional space. Unlike Cartesian coordinates, which use x, y, and z axes, cylindrical coordinates use the radius, angle, and height to locate a point in space.

2. What is the purpose of using cylindrical coordinates in triple integrals?

Cylindrical coordinates are useful in triple integrals because they simplify the calculation of volume and surface area of curved objects. They also make it easier to integrate functions with cylindrical symmetry.

3. How do you convert a triple integral with Cartesian coordinates to cylindrical coordinates?

To convert a triple integral from Cartesian coordinates to cylindrical coordinates, the following substitutions can be made: x = rcosθ, y = rsinθ, and z = z. The limits of integration must also be adjusted accordingly.

4. What are the advantages of using cylindrical coordinates over spherical coordinates?

While both coordinate systems are useful in triple integrals, cylindrical coordinates are often preferred because they are easier to visualize and work with. Spherical coordinates can be more complex and require more calculations.

5. Can cylindrical coordinates be used for any type of shape in a triple integral?

Yes, cylindrical coordinates can be used for any shape that has cylindrical symmetry, meaning the shape is symmetrical around a central axis. This includes objects such as cylinders, cones, and spheres.

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