Problem 1: A particle moves along the x-axis so that its acceleration at any time t is given by a(t) = 6t - 18. At time t = 0 the velocity of the particle is v(0) = 24, and at time t = 1 its position is x(1) = 20. (a) Write an expression for the velocity v(t) of the particle at any time t. (b) For what values of t is the particle at rest? (c) Write an expression for the position x(t) of the particle at any time t. (d) Find the total distance traveled by the particle from t = 1 to t = 3.
Solution:
(a) v(t) = ∫ a(t) dt = ∫ (6t - 18) dt = 3t2 - 18t + C 24 = 3(0)2 - 18(0) + C 24 = C so v(t) = 3t2 - 18t + 24
(b) The particle is at rest when v(t) = 0. 3t2 - 18t + 24 = 0 t2 - 6t + 8 = 0 (t - 4)(t - 2) = 0 t = 4, 2 (c) x(t) = ∫ v(t) dt = ∫ (3t2 - 18t + 24) dt = t3 - 9t2 + 24t + C 20 = 13 - 9(1)2 + 24(1) + C 20 = 1 - 9 + 24 + C 20 = 16 + C 4 = Cso x(t) = t3 - 9t2 + 24t + 4
Calculating the Area of Any Shape
Although we do have standard methods to calculate the area of some known shapes,
like squares, rectangles, and circles, but Calculus allows us to do much more.
Trying to find the area of shapes like this would be very difficult if it wasn’t for calculus.
The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence
