Volume Calculation under a Plane

The video tutorial explains how to find the volume under a plane over a square region in the XY plane. It begins by defining the region R as a 2x2 square and visualizes the plane passing through three points A, B, and C. The tutorial then derives the equation of the plane using vectors and the cross product. Finally, it calculates the volume under the plane using double integrals, resulting in a volume of 40 cubic units.

What is the shape of the region R in the XY plane?

• A circle

• A triangle

• A rectangle

• A square

Which points does the plane pass through?

• Points A, B, and D

• Points A, B, and C

• Points B, C, and D

• Points A, C, and D

What is the significance of the points A, B, and C in the problem?

• They are irrelevant to the calculation

• They form the vertices of a triangle

• They are used to define the plane

• They are the midpoints of the region

What is the first step in determining the equation of the plane?

• Solving a quadratic equation

• Determining vectors in the plane

• Calculating the area of the region

• Finding the midpoint of the points

How is the normal vector to the plane found?

• By adding the vectors

• By taking the cross product of two vectors

• By dividing the vectors

• By subtracting the vectors

What is the equation of the plane derived in the video?

• z = x + 8y + 1

• z = x - 8y + 1

• z = x + y + 8

• z = 8x + y + 1

What is the order of integration used in the double integral?

• dxdz

• dxdy

• dydx

• dzdx

What is the result of integrating 2x with respect to x?

• 2x^2

• x^2

• x^3

• 2x^3

What is the final volume calculated under the plane?

• 60 cubic units

• 50 cubic units

• 40 cubic units

• 30 cubic units

What is the integral of 18 with respect to x?

• 18x^2

• 18x

• 9x^2

• 9x