Multivariable Calculus (Calc III) - Complete Semester Course - Examples of integrating with polar coordinates, Multivariable Calculus

In this lesson, we do three examples of double integrals in polar or cylindrical coordinates. Example 1: In this computation, we want to integral the function 𝑓(𝑥,𝑦)=𝑥𝑦 over the domain 𝐷 , which is a quarter annulus in the first quadrant. The radial part 𝑟 varies from 1 to 2, and the angular part 𝜃 varies from 0 to 𝜋/2. The integral is separated into a product of two integrals, one in 𝑟 and one in 𝜃, and by using the substitution 𝑢=sin(𝜃), the result of the computation is 15/8. Example 2: We convert the integral to polar coordinates and integrate over 𝑟 and 𝜃, with 𝑟 going from 0 to 1 and 𝜃 from 0 to 2𝜋. The 𝑟 in the denominator and the 𝑟 from the differential 𝑑𝑟 cancel out, leading to a simple integral of 1 over the region, which results in 2𝜋. Example 3: We want to compute the volume inside the sphere 𝑥^2+𝑦^2+𝑧^2 = 16 and outside the cylinder 𝑥^2+𝑦^2 = 4. The process involves identifying the domain in polar coordinates, setting up the integral for the volume, and carrying out the integration to find the solution. In detail: 1. Draw a Picture: We start by visualizing the problem with a diagram. This helps us understand the spatial relationship between the cylinder 𝑥^2+𝑦^2 = 4 and the sphere 𝑥^2 + 𝑦^2 + 𝑧^2 = 16. The diagram shows the cylinder inside the sphere, and we're interested in the volume between these two shapes. 2. Find the Domain in 𝑟 and 𝜃: Next, we determine the domain for integration in polar coordinates. Since we're dealing with a circular shape on the xy-plane, it's natural to use polar coordinates, where 𝑟 is the radial distance from the origin and 𝜃 is the angle from the positive x-axis. We identify that 𝑟 ranges from the cylinder's radius of 2 to the sphere's radius of 4, and 𝜃 covers the full rotation around the circle, from 0 to 2𝜋. 3. Identify 𝑓(𝑥,𝑦): We set up the function 𝑓(𝑥,𝑦) using the equation of the sphere 𝑥^2+𝑦^2+𝑧^2 = 16. We need to express 𝑧 as a function of 𝑥 and 𝑦 to integrate over the xy-plane. In polar coordinates, this becomes 𝑓(𝑟cos𝜃,𝑟sin𝜃) = 16-sqrt(𝑟^2), which represents the positive half of the sphere above the xy-plane. 4. Setup and Solve the Integral: Finally, we set up the double integral to compute the volume. Since the volume is symmetrical about the xy-plane, we calculate the volume of the upper half and then double it. We integrate the function 𝑓(𝑟cos𝜃,𝑟sin𝜃) over the annular domain defined by the cylinder and sphere radii with respect to 𝑟 and 𝜃. We change variables to simplify the integral and solve it to find the volume enclosed between the sphere and the cylinder, getting 4𝜋/3⋅12^(3/2). #calculus #multivariablecalculus #mathematics #iitjammathematics #calculus3 #doubleintegrals #doubleintegration #polarcoordinates #cylindrical