Which of the following solids is not a solid of revolution?

The pyramid and cube do not have circular cross sections, so these are not solids of revolution. 2. When the region under a single graph is rotated about the x-axis, the cross sections of the solid perpendicular to the x-axis are circular disks.

How do you find volume in calculus?

In general, suppose y=f(x) is nonnegative and continuous on [a,b]. If the region bounded above by the graph of f, below by the x-axis, and on the sides by x=a and x=b is revolved about the x-axis, the volume V of the generated solid is given by V=∫abπ[f(x)]2dx.

What is the Shell method formula?

The shell method calculates the volume of the full solid of revolution by summing the volumes of these thin cylindrical shells as the thickness Δ x \Delta x Δx goes to 0 0 0 in the limit: V = ∫ d V = ∫ a b 2 π x y d x = ∫ a b 2 π x f ( x ) d x .

How do you find the volume of a solid?

Example 2 Determine the volume of the solid obtained by rotating the portion of the region bounded by y = 3√x y = x 3 and y = x 4 y = x 4 that lies in the first quadrant about the y-axis. First, let’s get a graph of the bounding region and a graph of the object.

How do you find the area of a solid disk?

In the case that we get a solid disk the area is, A = π(radius)2 A = π (radius) 2 where the radius will depend upon the function and the axis of rotation. In the case that we get a ring the area is,

How do you find the volume of a soup can?

To compute the volume of one shell, first consider the paper label on a soup can with radius and height . What is the area of this label? A simple way of determining this is to cut the label and lay it out flat, forming a rectangle with height and length . Thus the area is ; see Figure 6.3.2 (a).

How do you find the cross section of a solid object?

One of the easier methods for getting the cross-sectional area is to cut the object perpendicular to the axis of rotation. Doing this the cross section will be either a solid disk if the object is solid (as our above example is) or a ring if we’ve hollowed out a portion of the solid (we will see this eventually).