Final answer:
The question involves finding the area of an annular region using an iterated integral in polar coordinates by subtracting the area of the inner circle from the area of the outer circle.
Step-by-step explanation:
The student has asked how to use an iterated integral to find the area of the region bounded by the graphs of the equations x² + a² = y² + b². This equation represents a set of concentric circles centered at the origin with radii a and b. If a and b are constants, and if a is greater than b, the area of the annular region between these two circles can be found by subtracting the area of the smaller circle from that of the larger one. To calculate these areas using an iterated integral, the cartesian coordinates can be converted into polar coordinates since the symmetry of the problem suggests it.
To proceed, set up the integral in polar coordinates where r represents the radius and θ the angle. The area can be expressed as the double integral of r over the annular region:
- Convert the equation into polar coordinates as r² = a² - b² where a > b.
- Set the limits of integration for r from b to a, and for θ from 0 to 2π.
- Perform the double integral ∫ ∫ r dr dθ across these limits to find the area of the annular region.
The iterated integral will then provide the area of the region. This method of finding areas through integrals is a foundational concept in calculus, often represented graphically as the sum of infinitesimal strips or areas.