Double integral in polar form: how do you find the boundaries?

In summary, to find the area of the region bounded by r = 3 + 3sinQ, where Q = theta, a double integral is needed. The inner integral has bounds of 0 to 3 + 3sinQ, while the outer integral is a typical single-variable polar integral and has bounds of 0 to pi. The behavior of r repeats itself at 2pi and the value of sinQ becomes zero again at pi, allowing for the use of symmetry to double the value obtained from integrating from 0 to pi/2. A polar graph may be helpful in understanding the periodic nature of sine.
  • #1
winbacker
13
0
Hi I need to use a double integral to find the area of the region bounded by:

r = 3 + 3sinQ where Q = theta.

I know the bounds of the inner integral are from 0 to 3 + 3sinQ.

However, I do not know how to determine the bounds of the outer integral.

Any help would be greatly appreciated.
 
Physics news on Phys.org
  • #2
The outer integral is just a typical single-variable polar integral. Play with the equation to figure out when the behavior of r begins to repeat itself as theta varies, paying particular attention to the periodic nature of sine. Perhaps a polar graph might help.
 
Last edited:
  • #3
Ok, well I know that once theta = 2pi, the behavior of of sinQ will repeat itself. Should I then plug in 2pi to the equation and work with that?

I know the value of sinQ becomes zero again at pi. Does this mean the outer boundary is from 0 to pi?
 
  • #4
0 to pi, although you can exploit the symmetry of the figure by doubling the value you get integrating from 0 to pi/2.
 

1. What is a double integral in polar form?

A double integral in polar form is a type of integral that is used to calculate the area under a curve in polar coordinates. It is represented by the symbol ∫∫R where R represents the region of integration in the polar plane.

2. How do you convert a double integral in Cartesian form to polar form?

To convert a double integral in Cartesian form to polar form, you need to use the Jacobian transformation. This involves substituting x and y with their corresponding polar coordinates, r and θ, and multiplying the integrand by r. The boundaries of the integral also need to be converted to polar coordinates.

3. How do you determine the boundaries for a double integral in polar form?

The boundaries for a double integral in polar form depend on the shape of the region being integrated. In general, the inner boundary is determined by the smallest value of r, while the outer boundary is determined by the largest value of r. The angle θ also plays a role in determining the boundaries, and it can be helpful to sketch the region in the polar plane to visualize the boundaries.

4. Can you use a double integral in polar form to calculate volume?

Yes, a double integral in polar form can be used to calculate volume. This is done by integrating a function representing the cross-sectional area of a solid over a certain region in the polar plane. This is known as a triple integral in polar form.

5. What are some common applications of double integrals in polar form?

Double integrals in polar form are commonly used in physics and engineering to calculate moments of inertia and centers of mass for objects with rotational symmetry. They are also used in mathematics to find the areas of regions bounded by polar curves and to solve problems involving polar coordinates, such as finding the distance between two points in the polar plane.

Similar threads

Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
241
Confused about polar integrals and setting up bounds
    • Calculus and Beyond Homework Help
Replies
4
Views
744
How can I calculate the cumulative mass of a disk using disk mass density?
    • Calculus and Beyond Homework Help
Replies
25
Views
1K
Please evaluate this double integral over rectangular bounds
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
Double Integral via Appropriate Change of Variables
    • Calculus and Beyond Homework Help
Replies
3
Views
911
Find the bounds after changing the variables in a double integral
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Change of Variables: Integrals w/Polar Coordinates
    • Calculus and Beyond Homework Help
Replies
2
Views
848
Jacobian: how to change limits of integration?
    • Calculus and Beyond Homework Help
Replies
16
Views
1K
Symmetry of an Integral of a Dot product
    • Calculus and Beyond Homework Help
Replies
9
Views
947
Converting Cartesian to Polar (Double Integral)
    • Calculus and Beyond Homework Help
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top