Cross Sections and Solids of Revolution

Front

What is a solid of revolution?

Back

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional shape around a straight line (axis).

Front

What is the axis of rotation in solids of revolution?

Back

The axis of rotation is the fixed line around which the two-dimensional shape is rotated to form the solid.

Front

What is a cross section?

Back

A cross section is the intersection of a solid with a plane, resulting in a two-dimensional shape that represents a slice of the solid.

Front

How do you identify the shape of a cross section in a solid of revolution?

Back

The shape of a cross section in a solid of revolution depends on the position of the slicing plane relative to the axis of rotation.

Front

What is the formula for the volume of a solid of revolution using the disk method?

Back

The volume V is given by V = π∫[a to b] (f(x))^2 dx, where f(x) is the function being revolved.

Front

What is the formula for the volume of a solid of revolution using the washer method?

Back

The volume V is given by V = π∫[a to b] ((R(x))^2 - (r(x))^2) dx, where R(x) is the outer radius and r(x) is the inner radius.

Front

What is the difference between the disk method and the washer method?

Back

The disk method is used when there is no hole in the solid, while the washer method is used when there is a hole, requiring inner and outer radii.

Front

What is the significance of the fixed line in solids of revolution?

Back

The fixed line determines the shape and dimensions of the solid formed when the region is revolved around it.

Front

What types of shapes can be formed by revolving a rectangle around an axis?

Back

Revolving a rectangle around an axis can form a cylinder.

Front

What types of shapes can be formed by revolving a semicircle around an axis?

Back

Revolving a semicircle around an axis forms a sphere.