Let T be the region in the first quadrant bounded by the graphs of f(x)= cos x and g(x)= e^-2x

Question

Let T be the region in the first quadrant bounded by the graphs of f(x)= cos x and g(x)= e^-2x

(a) Find the area of T.

(b) Region T is the base of a solid. For this solid, the cross-section perpendicular to the x-axis are semi-circles with diameters extending from y=f(x) to y=g(x). Find the volume of this solid.

Answer 1

To get limits of integration wesolve equation f(x) = g(x); cos x = e^-x; x = 0 and x = 1.5232538

Second zero we get using TI-84.

(a) Area A = ∫₀^1.5232538(cosx - e^-2x)dx = (sinx + 1/2e^-2x)₀^1.5232538 = (sin1.5232538 + 1/2 e^-3.0465076) - (1/2) =

0.5226;

(b) V = π/8∫₀^1.5232538(cosx - e^-2x)²dx = 0.085458.

This integral we used TI-84 fnInt function from MATH menue

Answer 2

Area = ∫₀^a (cosx - e^-2x)dx , where a is the x-coordinate of the pt. of intersection (~ 1.523)

The radii of the semicircles are 1/2 of the vertical distance between the curves (ie 1/2 of the integrand above).

Since A_semicircle = 1/2πr² , Volume = ∫₀^a 1/2π · ((cosx - e^-2x)/2)²dx