Multivariable Calculus (Calc III) - Complete Semester Course - Example: double integral over a disk in polar coordinates, Multivariable Calculus

Let's calculate the volume between a circular region in the xy-plane and the surface z = 3x. The domain of integration is a disk centered at (5, 0) with radius 5, described by the equation (x - 5)^2 + y^2 = 25. We first set up a double integral in polar coordinates to find the volume, and then at the end, we revisit the setup for Cartesian coordinates. To establish our bounds: 1. Angular Bound (theta): We begin at -pi / 2 on the negative y-axis and rotate counterclockwise to pi / 2 on the positive y-axis, providing a continuous sweep of the region. 2. Radial Bound (r): We set the radial coordinate to range from 0 at the origin to the edge of the circle. We convert the circle's equation to polar form to get the appropriate upper bound for r. In closing, we compare this polar approach to alternative setups using Cartesian coordinates. Setting up the double integral in Cartesian form either as dy dx or dx dy would involve more complex integrands and ultimately trigonometric substitution. The polar setup is notably more efficient and elegant, making it the optimal choice for this problem. #mathematics #math #calculus3 #calculus #DoubleIntegral #PolarCoordinates #MultivariableCalculus #TrigonometricIntegration #iitjammathematics #cylindricalcoordinates #doubleintegration