Surface Area of a Plane Inside a Cylinder

The video tutorial explains how to find the surface area of a plane section within a cylinder using calculus. It begins by introducing the problem and visualizing it graphically. The tutorial then sets up a double integral to calculate the surface area, explaining the need for partial derivatives. It converts the integral from rectangular to polar coordinates for easier evaluation, and finally, it evaluates the integral to find the exact and approximate surface area values.

What is the shape of the region of integration on the XY plane?

• A circle centered at the origin

• A square centered at the origin

• An ellipse centered at the origin

• A rectangle centered at the origin

What is the radius of the circle that represents the region of integration?

• 1

• 4

• 2

• 3

What is the partial derivative of the function with respect to X?

• 4

• 2

• 1

• 3

What is the partial derivative of the function with respect to Y?

• 4

• 1

• 2

• 3

What is the value of the integrand function in polar coordinates?

• Square root of 21

• Square root of 25

• Square root of 16

• Square root of 9

What are the limits of integration for R in polar coordinates?

• 0 to 2

• 0 to 3

• 0 to 4

• 0 to 1

What are the limits of integration for theta in polar coordinates?

• 0 to 4 pi

• 0 to 3 pi

• 0 to 2 pi

• 0 to pi

What is the exact surface area of the plane inside the cylinder?

• 4 pi square root 21

• 3 pi square root 21

• 2 pi square root 21

• 5 pi square root 21

What is the approximate surface area of the plane inside the cylinder?

• 77.5863 square units

• 67.5863 square units

• 47.5863 square units

• 57.5863 square units

What shape does the surface area of the plane inside the cylinder resemble?

• A circular region

• A rectangular region

• A square region

• An elliptical region