Multivariable Calculus (Calc III) - Complete Semester Course - Average value with polar coordinates example, Multivariable Calculus

Let's compute the average f(x,y)=xy over the quarter over radius 5 in the xy-plane. We will use polar coordinates to set up and evaluate the double integrals. We start by sketching our domain of integration, which appears as the quarter of a solid disk in this quadrant. To determine the average value of f, we calculate it as the function integrated over this domain D, divided by the area of D. This area can be computed by double integrating the value 1 over the domain. For easier computation, we switch to polar coordinates where x=rcos⁡(θ) and y=rsin⁡(θ), with radial bounds from 0 to 5 and angular bounds from 0 to pi/2​. We then express xy in polar coordinates and introduce the extra r from the Jacobian in the integration. The function simplifies into a product of functions dependent on r and θ, allowing us to factor the integral into two separate single-variable integrals. For the area of D, while it's straightforward from geometry as 25π/4​, we also demonstrate this by integration. After setting up and solving the integrals, we perform a u-substitution for the angular part and evaluate a straightforward power integral for the radial part. We finish by calculating the necessary fractions and reductions to arrive at the final average value of the function over our domain. #mathematics #math #multivariablecalculus #averagevalue #iitjammathematics #calculus3 #applicationsofintegration #doubleintegrals #doubleintegration #polarcoordinates #mathtutorial