Volume Integrals and Cross-Sections

This video tutorial explains how to determine the volume of a solid using the method of slices, integrating with respect to y. The base of the solid is defined by two curves, and the cross sections are semi-circles perpendicular to the y-axis. The tutorial covers setting up the integral, calculating the radius of the semi-circles, finding the antiderivative, and computing the final volume. The process involves algebraic verification of limits and detailed steps to solve the integral, resulting in the volume of the solid.

What is the shape of each cross-section in the volume described?

• Triangle

• Square

• Semi-circle

• Circle

Which axis are the cross-sections perpendicular to?

• z-axis

• None of the above

• y-axis

• x-axis

What are the limits of integration for the volume integral?

• x = 0 to x = 10

• y = 7 to y = 13

• x = 7 to x = 13

• y = 0 to y = 10

How is the radius of each semi-circle determined?

• The full length of x sub r plus x sub l

• The full length of x sub r minus x sub l

• Half the length of x sub r minus x sub l

• Half the length of x sub r plus x sub l

What is the integrand function for the volume integral?

• pi times the square of the radius

• pi times the radius

• One-half pi times the square of the radius

• One-half pi times the radius

What is the first step in finding the antiderivative?

• Factor out pi

• Find the derivative

• Integrate directly

• Square the expression

What is the result of the integration process?

• A constant value

• A polynomial function

• An exponential function

• A trigonometric function

What is the final volume of the solid?

• Both A and B

• 407.1504

• Neither A nor B

• 648 fifths pi

What is the approximate numerical value of the volume?

• 300.5678

• 200.4321

• 407.1504

• 500.1234

What mathematical operation is used to find the volume?

• Integration

• Multiplication

• Division

• Differentiation