1/29/11

Alright well at the beginning of this week, we finished up with the UNO study learning how to revolve a graph about the x-axis using the washer and disk methods. This actually gives you the volume of a certain solid based off of the equation you're given in the problem. Then later in the week we went over position, velocity, and acceleration functions and how to apply integrals with them. For instance, displacement (which is the same thing as position) is the integral of velocity; and distance is the absolute value of the integral of velocity.
Here's an example:

1.) Meg accelerates her car, giving it a velocity of v = t^1/2 - 2, in feet per second, at time t, in seconds after she started accelerating.
A: Find the time(s) at which v = 0
*All they're asking for here is "t", so all you do is plug in 0 for v (i.e.-set the equation they gave you equal to zero and solve for t) So you should get this:
t^1/2 - 2 = 0
t^1/2 = 2 ...(*t^1/2 is the same thing as the squareroot of t, so to get rid of that all you do is square both sides of the equation)
(t^1/2)^2 = 2^2
t=4
B: Find her net displacement for the time interval [1,9].
*Soooo, they give you the time interval [1,9]..which is going to be your bounds for your integral
*Since they're asking for the displacement, you're going to simply take the integral of the velocity function with bounds 1,9
*So plugging into that formula you should get this:
S1,9 t^1/2-2 dt
Now taking the integral of that equation you should end up with:
2/3t^3/2 - 2t
Now use the Fundamental Theorem of Calculus to plug in 9 and 1 like this:
2/3(9)^3/2 - 2(9) - [2/3(1)^3/2 - 2(1)]
= 4/3 ft
is displacement
C: Find the total distance she travels for the time interval [1,9].
*So now since they're asking for distance, all we're going to do is use the distance formula..which says to take the absolute value of the integral of velocity
*So plugging that into your calculator with bounds [1,9] you should end up with 4
*So 4 ft is your total distance

*Again this week was a pretty good week because I understood everything :) haha....All except for #13 on the homework worksheet we had Thursday night..Oh and also, I have no clue how to integrate secant..or anything besides sine and cosine ha