Other Applications of Integration

Question

1. The area bounded by the curve y = e^−x, the axes and the line x = 2 is revolved about ox. Find the volume generated. (use circular disk method)

2. Find the volume of the torus formed by revolving the circle x² + y² = a² about the line x = b(b > a). (Use circular ring method)

Answer 1

Sketch:

At any random spot between x = 0 and x = 2 the cross section of the solid is a circle. The radius of that circle is the function value of y = e^-x so the area of that circle is A = π(r^π2) = π(e^-x)² = πe^-2x so the volume is:

V(x) = ₀∫² πe^-2x dx = π•(-1/2)e^-2x evaluated between 0 and 2 = (-π/2)(e^-4) - -π/2(1) = π/2 - π/(2e⁴) ≈ 1.542 cubic units

For the second problem, you have a circle rotated around a center line giving a doughnut shape. Integrating in the y-direction provides a "washer" for each cross section. Calculate the inside radius as a function of "y" and then the outside radius as a function of "y", then use the fact that the area of a "washer" is π(r_o² - r_i²). Hint: The inside radius at y = 0 is b - a and the inside radius at y = a is b while the outside radius at y = 0 is a + b and the outside radius at y = a is b. The Volume is 2•₀∫^a A_washer dy