Barrons AP Calculus - David Bock

when t = 0, the particle is at (1, 0) with

(a) Find the position vector R at any time t.

(b) Find a Cartesian equation for the path of the particle, and identify the conic on which P

moves.

SOLUTIONS:

(a) and since when t = 0, it follows that c 1 = c 2 = 0. So

Also and since when t = 0, we see that c 3 =

c 4 = 0. Finally, then,

(b) From (a) the parametric equations of motion are

x = cos 2t, y = 2 sin t.

By a trigonometric identity,

P travels in a counterclockwise direction along part of a parabola that has its vertex at (1, 0) and

opens to the left. The path of the particle is sketched in Figure N8–1; note that −1 ≤ x ≤ 1, −2 ≤ y ≤

2.

FIGURE N8–1

C. OTHER APPLICATIONS OF RIEMANN SUMS

We will continue to set up Riemann sums to calculate a variety of quantities using definite integrals.
In many of these examples, we will partition into n equal subintervals a given interval (or region or
ring or solid or the like), approximate the quantity over each small subinterval (and assume it is
constant there), then add up all these small quantities. Finally, as n → ∞ we will replace the sum by