Use cylindrical coordinates to find the volume of the solid. The solid that is bounded above by

Use cylindrical coordinates to find the volume of the solid....

Key Concepts

Triple Integration for Volume

Triple integration involves setting up an integral to compute the volume of a solid by integrating a constant (usually 1) over the region of interest. In cylindrical coordinates, this process requires properly identifying the limits for r, ?, and z based on the bounding surfaces, and then integrating in the chosen order (typically z, then r, then ?) while including the Jacobian factor. This method is especially useful when the boundaries of the solid fit the symmetry inherent to cylindrical coordinates.

Cylindrical Coordinates

Cylindrical coordinates provide a system of representing points in space using a radius, an angle, and a height (r, ?, z). This coordinate system is particularly useful for problems that exhibit cylindrical or rotational symmetry. In this system, x and y are expressed in terms of r and ? (x = r cos?, y = r sin?), while z remains unchanged, simplifying the integration setup for solids with circular cross-sections or symmetry around the z-axis.

Jacobian Determinant

When transforming from Cartesian to cylindrical coordinates, the volume element changes. The Jacobian determinant, which in cylindrical coordinates is the factor r, must be included in the integrand. This factor accounts for the stretching of the coordinate grid and ensures that the integration correctly measures volumes in the new coordinate system.

Intersection of Surfaces

Determining the intersection of surfaces is vital for setting the limits of integration. By equating the equations of the sphere and the cone (expressed in cylindrical coordinates), one can find the curve along which these surfaces meet. This intersection provides critical bounds for the integration variables, ensuring that the integration encompasses the exact region defined by the solid's boundaries.

Transcript

00:01 Okay, we want to use cylindrical coordinates to find the volume that is bounded below by the cone and above by the sphere.

00:11 Okay, that cone is the 45 degree angle cone because, for example, if y was zero then z is plus minus x.

00:21 Okay, so we have this cone out here that's at 45 degrees in both the x and y direction and then we have this sphere on top of radius one.

00:36 All right, so z is going from the cone up to the sphere.

00:48 So let's go ahead and switch to cylindrical here.

00:52 So that's the square root of r squared which is r and z is the square root of one minus x squared minus y squared or the square root of one minus r squared.

01:08 Okay, then when we look down here in two dimensions, this is x, this is y, we have a circle and its radius is one.

01:26 Um, z squared, oh no, i almost made a mistake there.

01:30 I need to solve this system simultaneously.

01:33 I'm going to square the bottom one so i have z squared equals x squared plus y squared.

01:40 I'm going to plug it in right here so i get x squared plus y squared plus x squared plus y squared equals one.

01:52 So that's r squared plus r squared equals one.

01:55 Two r squared equals one.

01:58 R squared equals one half.

02:08 So r is one over the square root of two.

02:15 Okay, so this is r equals one over the square root of two and then we're going to go all the way around.

02:20 So r is going from zero to r equals one over the square root of two and then theta is going from zero to two pi...

Please give Ace some feedback