Intersection of two cylinders

The image on the left shows two intersecting cylinders. The vertical cylinder is

x 2 + y2 = 1

and the horizontal cylinder is

x2 + z2 = 1.

The image on the right shows the set of all points that lie on or inside both cylinders.

Figure 1.

Figure 2.

With these two applets you can rotate the intersection of the two cylinders to get a better idea of the actual shape of this region. Both applets begin with the viewpoint on the x-axis. The first applet spins the region about the z-axis. The second applet rotates the region 90 degress and back around the y-axis.

Calculating the volume (Method 1)

Looking at the region of intersection of these two cylinders from a point near the z-axis, as in Figure 2 above, we see that the sides of this region lie on the cylinder

x 2 + y2 = 1

and the top and bottom of the region lie on the cylinder

2 2 x + z = 1.

We can characterize the region as the set of all points satisfying

2 2 V~ -----2 V~ -----2 x + y < 1, - 1 - x < z < 1- x .

The volume of the region is

integral integral V~ -2- integral 1 1- x V~ -----2 1 2 16- V = -1 - V~ 1-x2 2 1- x dydx = -14|\1 - x /|dx = 3 .

Calculating the volume (Method 2)

Looking at the region of intersection of these two cylinders from a point on the x-axis, we see that the region lies above and below the square in the yz-plane with vertices at (1,1), (-1,1), (-1,-1), and (1,-1). The diagonals of this square divide it into 4 regions, labelled I, II, III, and IV. Over the triangular regions I and III the top and bottom of our solid is the cylinder

x2 + y2 = 1.

Over the triangular regions II and IV the top and bottom of our solid is the cylinder

2 2 x + z = 1.

Using symmetry, the volume of the region of intersection is

integral 1 integral y V~ ---- integral 1 V~ ------ V = 8 1 - y2dz dy = 16 y 1- y2dy. 0 -y 0