Cylindrical coordinates, finding volume of solid

• ohlala191785
In summary, the problem asks for the volume of a solid created by a cylinder with radius a and height r, intersecting a sphere of radius a centered at the origin. The integrand is 1 and the limits of integration for z are -√a2-r2 and +√a2-r2. The limits of r are from 0 to a, and the limits of θ depend on r and can be found using a sketch of the r,θ-plane or by using cartesian coordinates. The cylinder is only defined for x>0.
ohlala191785

Homework Statement

Find the volume of the solid that the cylinder r = acosθ cuts out of the sphere of radius a centered at the origin.

Homework Equations

Cylindrical coordinates: x = rcosθ, y = rsinθ, z=z, r2 = x2+y2, tanθ = y/x

The Attempt at a Solution

So I know that the equation for the sphere is x2+y2+z2=a2 since it's centered at the origin and has a radius of a. And I'm pretty sure the integrand is 1, so the integral should look like ∫∫∫rdzdrdθ. For the limits of z, I solved for z in the sphere equation and got z=±√a2-x2-y2, which in cylindrical coordinates is z=±√a2-r2. Therefore the limits of integration for z are -√a2-r2 and +√a2-r2.

The limits of r should be from 0 to a since a is the radius of the sphere. But I have no idea what the limits of θ are. Is it from 0 to 2π because it is a sphere?

The limits of your cylinder are different, and depend on r. You can draw a sketch of the r,θ-plane to see this.
If cylindrical coordinates do not work, try cartesian coordinates. The integration limits are easier there.

You can state your problem as a double integral, where the integrand is a function of the surface at point (x,y) - or (r,θ). Can you see that function?
Beware: as stated, the cylinder is defined only for x>0 (where cosθ>0).

1. What are cylindrical coordinates?

Cylindrical coordinates are a 3-dimensional coordinate system that is commonly used in mathematics, physics, and engineering. They consist of a radial distance from the origin, an angle in the xy-plane, and a height or z-coordinate.

2. How do you convert from rectangular coordinates to cylindrical coordinates?

To convert from rectangular coordinates (x,y,z) to cylindrical coordinates (r,θ,z), you can use the following equations:
r = √(x² + y²)
θ = tan⁻¹(y/x)
z = z

3. How do you find the volume of a solid using cylindrical coordinates?

To find the volume of a solid using cylindrical coordinates, you can use the following formula:
V = ∫∫∫ r dr dθ dz
This involves integrating over the specified region and can be solved using techniques from multivariable calculus.

4. Can you use cylindrical coordinates to find the volume of a non-cylindrical shape?

Yes, cylindrical coordinates can be used to find the volume of a non-cylindrical shape as long as the shape can be expressed in terms of cylindrical coordinates. This may involve breaking the shape into smaller cylindrical sections and integrating over each section.

5. What are some real-world applications of cylindrical coordinates?

Cylindrical coordinates have many practical applications in fields such as engineering, physics, and astronomy. They are commonly used to describe the motion of objects in circular paths, such as the orbit of planets or satellites. They are also used in the analysis and design of cylindrical structures, such as pipes and cylinders. Additionally, they are used in fluid dynamics to model the flow of liquids and gases in cylindrical containers or pipes.

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