Surface Area of a Plane Inside a Cylinder

The video tutorial explains how to find the surface area of a plane within a cylinder using calculus. It begins with a graphical representation of the problem, showing the circle of integration on the xy-plane. The tutorial then sets up a double integral to calculate the surface area, explaining the need to find partial derivatives with respect to x and y. The integral is converted to polar coordinates for easier evaluation, and the limits of integration are defined. Finally, the integral is evaluated to find the exact and approximate surface area, with a final review of the graphical representation.

What is the radius of the circle formed by the cylinder on the xy-plane?

• 3

• 2

• 1

• 4

What is the function f(x, y) representing the plane?

• 4x + 3y

• x^2 + y^2

• 2x + 3y

• 3 + 2x + 4y

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

• 2

• 3

• 4

• 0

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

• 3

• 0

• 2

• 4

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

• 20

• √20

• 21

• √21

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

• 0 to 3

• 0 to 4

• 0 to 2

• 0 to 1

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

• 0 to π

• 0 to 2π

• 0 to 3π

• 0 to π/2

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

• π√21

• 8π√21

• 4π√21

• 2π√21

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

• 60.7890 square units

• 50.1234 square units

• 55.4321 square units

• 57.5863 square units

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

• Triangle

• Square

• Ellipse

• Circle