The Method of Cylindrical Shells Help

Example 1

Use the method of cylindrical shells to calculate the volume of the solid enclosed when the curve y = x² , 1 ≤ x ≤ 3, is rotated about the y -axis.

Solution 1

As usual, we think of the region under y = x² and above the x -axis as composed of vertical segments or strips. The segment at position x has height x². Thus, in this instance, h = x², r = x , and the volume of the cylinder is 2πx · x² · Δ x . As a result, the requested volume is

We easily calculate this to equal

Example 2

Use the method of cylindrical shells to calculate the volume enclosed when the curve y = x ² , 0 ≤ x ≤ 3, is rotated about the x -axis ( Fig. 8.20 ).

Fig. 8.20

Solution 2

We reverse, in our analysis, the roles of the x - and y -axes. Of course y ranges from 0 to 9. For each position y in that range, there is a segment stretching from to x = 3. Thus it has length . Then the cylinder generated when this segment (thickened to a strip of width Δ y ) is rotated about the x -axis has volume

The aggregate volume is then

Use the method of washers to calculate the volume of the solid enclosed when the curve , is rotated about the line y = −1. See Fig. 8.21.

Fig. 8.21

Solution 1

The key is to notice that, at position x , the segment to be rotated has height the distance from the point on the curve to the line y = −1. Thus the disk generated has area . The resulting aggregate volume is

Calculate the volume of the solid enclosed when the area between the curves x = ( y − 2)² + 1 and x = −( y − 2)² + 9 is rotated about the line y = −2.

Fig. 8.22

Solution 2

Solving the equations simultaneously, we find that the points of intersection are (5, 0) and (5, 4). The region between the two curves is illustrated in Fig. 8.22.

At height y , the horizontal segment that is to be rotated stretches from (( y − 2)² + 1, y ) to (−( y − 2)² + 9, y ). Thus the cylindrical shell that is generated has radius y − 2, height 8 − 2( y − 2)² , and thickness Δ y . It therefore generates the element of volume given by

2 · ( y − 2) · [8 − 2( y − 2)² ] · Δ y.

The aggregate volume that we seek is therefore