Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the graphs of y=x, y=7-x, and the x-axis about the x-axis.

Answer 1

The "region bounded by the graphs" is an isoceles right triangle whose three vertices are the points (0, 0), (3.5, 3.5), and (7,0). The right angle is at the point (3.5, 3.5) at the top of the triangle, and the hypotenuse is the line that connects the points (0,0) and (7,0). The two legs are diagonal line segments at 45 degrees. Draw the triangle. The triangle's height is 3.5; in other words, y goes from 0 to 3.5.

We're going to rotate this triangle around the X axis. Think of the triangle as composed of extremely thin, horizontal strips. Let y be the hight above the X axis of a typical horizontal strip. Then the length of the strip is 7-2y. For example, the lowest strip (with y=0) is of length 7 (it runs along the hypotenuse), and the shortest strip (with y=3.5) is of length 0 (it's at the right angle up at the top of the triangle).

As our typical horizontal strip gets rotated around the X axis, it sweeps out a cylinder whose wall is extremely thin (because the strip is extremely thin). The radius of the cylinder is y (so the circumference of the cylinder is 2πy), and the hight of the cylinder is 7-2y (because that's the length of the strip). Of course, the cylinder is lying horizontally (the axis of the cylinder is the X axis).

What is the volume of this cylinder? Imagine that the cylinder was manufactured by rolling an extremely thin rectangular plate into a cylinder and welding the seam. Then the volume of the cylinder is approximately equal to the volume of the plate. The dimensions of the extremely thin plate are 2πy (because that's the circumference of the cylinder) by 7-2y (because that's the height of the cylinder) by dy (because that's our name for the length of the "extremely thin" quantity).

So the volume of the rectangular plate is (2πy)(7-2y)dy.

Therefore the volume of the cylinder is (2πy)(7-2y)dy.

As y goes from 0 to 3.5 (see the last sentence in the first paragraph), the sum of the volumes of all the cylinders is therefore

∫ (from y=0 to y=3.5) (2πy)(7-2y)dy

= 2π·∫ (from y=0 to y=3.5) y(7-2y)dy

= 2π·∫ (from y=0 to y=3.5) (-2y² + 7y)dy

Now take the antiderivative:

= 2π (((-2/3)y³ + (7/2)y²) (from y=0 to y=3.5))

When you evaluate this, write the 3.5 as 7/2 because no one likes to multiply decimal points.

= 2π((-2/3)(7/2)³ + (7/2)(7/2)²)

= 343π/12

Let's check this result by dividing our triangle into extremely thin vertical strips. We'll measure the volume swept out by half of our triangle, from x=0 to x=3.5, and then double the result. The height of a typical vertical strip is x and its width is dx. When rotated around the X axis, the strip sweeps out a disk whose radius is x and whose thickness is dx. The volume of this disk is therefore πx²dx and the volume of all the disks as x goes from 0 to 3.5 is

∫ (from x=0 to x=3.5) πx²dx

= π·∫ (from x=0 to x=3.5) x²dx

Now take the antiderivative:

= π((x³/3) (from x=0 to x=3.5)

When you evaluate this, write the 3.5 as 7/2 because no one likes to multiply decimal points.

= π(((7/2)³)/3)

= 343π/24

When we double this result, we get 343π/12, which agrees with our volume from the cylindrical shell method.

Hope this helps.

Answer 2

y = 7 - y; y = 3.5;

Volume v = 2π∫₀^3.5(7 - 2y)ydy = 2π(7y²/2 - 2y³/3)₀^3.5 = 343π/12