Calculus 1 Notes: Disk Method

Calculus 1 Notes: Disk Method

· In order to understand the disk method, you must first examine the idea of a cross-section. State the shape of the cross-sections of several items in the room.

· To further your understanding, you need to be able to visualize a curve rotating about a line. For example, what 3-D shape would the following curve create if rotated about the x-axis?

What about this shape when rotated about the y-axis?

· Take a look at a few other curves and try to imagine what 3-D figure would be created if they were spun or rotated about a given axis of revolution.

· Problem 1: Find the volume of the solid of revolution formed by rotating the graph of about the x-axis when

First, we must be able to graph and visualize the problem:

We also need to identify the cross-sections: ____________

Develop a formula for the volume of this region:

Whenever you rotate a region around an axis of symmetry and the cross-sections are disks, the formula for the volume of that region is:

· Problem 2: Find the volume of the region bounded by: when that region is rotated about the x-axis.

· Problem 3: Rotate the area bounded by and the x-axis about the x-axis and calculate the volume that results. (hint: you need to figure out where the curve intercepts the x-axis—to get limits).

What if we were to rotate about a line other than the x-axis?

· Problem 4: Find the volume of the region obtained by rotating the area enclosed by: about the line .

· Problem 5: Consider the region bounded by and rotated about

· Problem 6: We can also rotate about the y-axis. The concept is the same except we have a horizontal representative rectangle. Consider the region bounded by ; . Find the area that results when rotating about the y-axis.

(positive sq. root so Pos.x-values only)