Calculating the Area of the Region Inside r9sinθ but Outside r2

When dealing with areas in polar coordinates, one common problem is calculating the region enclosed by two curves. In this article, we will explore in detail how to find the area of the region inside the curve r 9sinθ but outside the circle r 2.

Step 1: Identifying the Curves

The polar curve r 9sinθ represents a circle centered at (0, 4.5) with a radius of 4.5. On the other hand, the circle r 2 is centered at the origin with a radius of 2.

Step 2: Finding the Points of Intersection

To find where the two curves intersect, we set them equal to each other:

r 9sinθ 2

Solving for sinθ, we get:

sinθ 2/9

Now we find the angles θ where this occurs:

θ arcsin(2/9) and θ π - arcsin(2/9)

Step 3: Setting Up the Area Integral

The area A of the region between two polar curves from θ a to θ b is given by:

A 1/2 ∫ab (router2 - rinner2) dθ

Here router 9sinθ (the larger radius), and rinner 2 (the smaller radius).

The limits of integration will be a arcsin(2/9) and b π - arcsin(2/9).

Step 4: Calculating the Area

The integral can be computed as:

A 1/2 ∫arcsin(2/9)π - arcsin(2/9) (81sin2θ - 4) dθ

Simplifying the integrand:

A 1/2 ∫arcsin(2/9)π - arcsin(2/9) (81sin2θ - 4) dθ

The integral of sin2θ can be computed using the identity:

sin2θ (1 - cos2θ)/2

Thus:

∫ sin2θ dθ (θ/2) - (sin2θ)/4 C

Substituting back, the integral becomes:

A 1/2 [81 ∫ sin2θ dθ - 4 ∫ dθ]

A 1/2 [81 ((θ/2) - (sin2θ)/4) - 4θ] evaluated from arcsin(2/9) to π - arcsin(2/9)

Evaluated numerically, the final area can be computed, but the setup gives you the complete method to find the area of the desired region. For a numerical approximation, you could use tools like numerical integration methods or computational software to evaluate the definite integral.

Conclusion

Understanding how to calculate the area of a region in polar coordinates is essential in various fields of mathematics and physics. By following the steps outlined above, you can confidently solve problems like finding the area inside one polar curve but outside another. Utilizing computational software or numerical methods will help in obtaining an accurate numerical approximation.