Cylindrical Coordinates and Volume Calculations

This video tutorial covers the use of triple integrals in cylindrical coordinates to calculate the volume of solids. It begins with an introduction to cylindrical coordinates, explaining the representation of points in space using r, theta, and z. The tutorial then delves into the process of setting up and evaluating triple integrals, emphasizing the importance of integration limits. Two examples are provided: one involving a paraboloid and another involving a cylinder, demonstrating the practical application of these concepts in determining the volume of bounded regions.

What is the representation of a point in space using cylindrical coordinates?

• p, q, r

• x, y, z

• r, theta, z

• a, b, c

In cylindrical coordinates, what does the variable 'r' represent?

• The angle from the x-axis

• The height from the xy-plane

• The directed distance from the origin in the xy-plane

• The distance along the z-axis

What is the differential element 'dV' in cylindrical coordinates?

• r dz dr dtheta

• dr dtheta dz

• dx dy dz

• r dx dy dz

What is the upper limit of integration for 'z' in the first example?

• 9 - x^2 - y^2

• x^2 + y^2 - 9

• 9 - r^2

• r^2 - 9

In the first example, what shape is formed by the xy-trace?

• A circle

• A triangle

• A square

• An ellipse

What is the final volume of the solid in the first example?

• 81 pi/3

• 81 pi

• 81 pi/2

• 81 pi/4

In the second example, what is the equation of the surface bounding the solid from above?

• z = 4 - sqrt(x^2 + y^2)

• z = 4x^2 + 4y^2

• z = 4 + sqrt(x^2 + y^2)

• z = x^2 + y^2 - 4

What is the radius of the circle in the xy-trace for the second example?

• 2

• 1

• 4

• 3

What is the final volume of the solid in the second example?

• 10 pi/3

• 10 pi

• 10 pi/2

• 10 pi/4

In the second example, what is the lower limit of integration for 'z'?

• z = 1

• z = 0

• z = 4

• z = -1