I'm working on problem 19.39 of Mendelson's 3000 Solved Problems Of Calculus.
A particle is moving along a line with acceleration: $$ a = \sin(2t)+ t^2\ \mathrm{ft/s^2} . $$
At time $t=0$, the velocity is $3\ \mathrm{ft/s}$.
What is the distance between the particle's location at $t=0$ and its location at $t=\frac{\pi}{2}$?
I tried to solve the problem without doing the calculations, using a graphing calculator and " area functions" ( playing the role of integrals) but I do not get the right result which out to be : $$ \frac{\pi^4+288\pi}{192} $$ What I do not understand is that, according to Mendelson's solution , the distance travelled at t=0 is not null, but rather is equal to the constance C of the integral of the velocity function. ( This I understand : distance is the integral of velocity).
Why is the distance travelled at t=0 not null if, as I assume, t=0 is the beginning of the movement?
Is this assumption wrong?
Also, why does the " area function "apprach" not work?
Could you explain me what I am misssing?
Below, an image showing what I get with the graphing calculator
So, $a(t) = \sin(2t)+t^2$
By integrating with respect to $t$ we get:
$v(t) = \int \sin(2t)+t^2~dt = -\frac{\cos(2t)}{2}+\frac{t^3}{3}+C$ for $C$ some constant
Remembering our initial condition that $v(0)=3$, we solve for $C$ as being $3.5$
Now... we can look at the definite integral of $v$ in order to determine the distance between the starting position and the ending position. In particular, we are finding
$$\int\limits_0^{\frac{\pi}{2}}v(t)~dt = \int\limits_0^{\frac{\pi}{2}}\left( -\frac{\cos(2t)}{2}+\frac{t^3}{3}+3.5\right)~dt$$
$$=\left[-\frac{\sin(2t)}{4}+\frac{t^4}{12}+3.5t\right]^{t=\pi/2}_{t=0}$$
$$=\left(0+\frac{\pi^4}{12\cdot 2^4}+3.5\cdot \frac{\pi}{2}\right)-\left(0+0+0\right) = \frac{\pi^4+288\pi}{192}$$
Your error appears to have been using a definite integral when calculating $v(t)$ which would have given you the change in velocity over a time period rather than what the velocity was after a certain time. By using an indefinite integral and calculating for the coefficient $C$ by using the provided initial conditions, you should have then been able to continue the problem.