Multivariable Calculus (Calc III) - Complete Semester Course - Areas between circles with double integrals, Multivariable Calculus

In this video, we look at two examples: (1) Find the area enclosed between y^2=2x-x^2 and y^2=8x-x^2, and (2) find the (rightmost) area enclosed between x^2+y^2=9 and y^2=8x-x^2. The first can be solved with geometry, but we decide to express it as a double integral using polar coordinates for practice. This approach will also prepare us for a variation of the problem (the second example) where geometrical methods are less apparent. In the first example, we complete the square for both equations to describe the circles algebraically. The resulting equations represent circles centered at (1,0) with radius 1 and (4,0) with radius 4, matching our graphical observations. We draw rays from the origin through the region of interest to determine the radial (r) and angular (θ) bounds. The domain for θ covers the right half-plane from -pi/2 to pi/2. To find the radial bounds, we examine the entry and exit points of the rays through the region, corresponding to the smaller and larger circles respectively. Then we convert the circle equations to polar coordinates. For the smaller circle, y^2 = 2x - x^2, we obtain r = 2sin(θ), and for the larger circle, y^2 = 8x - x^2, we derive r = 8sin(θ). These provide the radial bounds for our double integral. To compute the area enclosed between the circles, we set up a double integral over the region D. We express the integral in polar coordinates, where the differential area element dA becomes r dr dθ. With r varying from 2sin(θ) to 8sin(θ) and θ from -pi/2 to pi/2, we integrate r first, giving us the area 15pi, consistent with the geometric solution. #mathematics #math #multivariablecalculus #doubleintegrals #doubleintegration #polarcoordinates #iitjammathematics #calculus3 #IntegrationTechniques #MathEducation #cylindrical