The region bounded by those curves is rather small! And the simplest way to find the volume rotated around the x-axis is to do it as two integrals. The volumes bounded above by y= sec(x), below by y= 0, on the left by x= -1, and on the right by x= 1, rotated around the x-axis, is . The volume bounded above by y= 1, below by y= 0, on the left by x= -1, and on the right by x= 1, rotated around the x-axis, is (actually the volume of a cylinder with radius 1 and height 2). The volume you want is the difference between those two. That is the same as .
That is based on the fact that the volume of a cylindrical "donut" with inner radius r, outer radius R, and height h, is which is the same as the volume of the whole cylinder, minus the volume of the inner cylinder, .