Triple Integration. Volume under a Paraboloid.

Use Polar Coordinates to find the Volume of the Solid below the Paraboloid z = 18-2x^2 -2y^2 and above the XY-Plane.

Double Integrals in Polar Coordinates.
0:00 Introduction. 0:38 Double Integrals in Polar Coordinates.

2:45 R = {r, theta; 1 less than or equal less than or equal 2, 0 less than or equal theta less than or equal pi }.

4:31 The concept of the limits of r and theta in Polar coordinates.

7:04 The area of a sector is 1/2 r to the 2nd theta.

10:54 delt A = r delta r delta theta. Or dA=r dr d theta.

11:48 Change to Polar Coordinates in a double Integral.

13:31 The Arc Length is r * theta.

14:41 Notes that you need for the Polar Coordinates.

15:04 Example: Evaluate the Double Integral of 3x+4y to the 2nd where R is the region in the upper half plane bounded by the circles x to the 2nd + y to the 2nd =1 and x to the 2nd + y to the 2nd = 4.

19:47 Example: Find the volume of the solid bounded by the plane z = 0 and the paraboloid z = 1- x to the 2nd- Y to the 2nd.

24:52 In General, the Area of a Sector when r1 and r2 are not constants but functions of theta.

25:49 Find the VOLUME of the solid under the Paraboloid z = x to the 2nd + y to the 2nd, above the XY-Plane, and inside the Cylinder x to the 2nd + y to the 2nd = 2x.

Calculate the iterated integral by converting it to polar coordinates.

The double integral of ( x+ y ) where x varies from y to the square root of (2 - y^2), and y varies from 0 to 1.