Calculus study realm

Δx = ∫v(t) dt

v(t) = ∫a dt = ∫3 cost dt = 3sint + C
v(0) = -3/2 = 3sin(0) + C; C = -3/2
Thus, v(t) = 3sint - 3/2

Displacement: Δx = ∫(0 to π/2) v(t) dt = ∫(0 to π/2) 3sint - 3/2 dt = (-3cost - 3/2t)|(0 to π/2) = -3cos(π/2) - 3/2(π/2) + 3cos(0) + 3/2(0) = - 3π/4 + 3 = 0.64

Distance traveled: d = ∫(0 to π/2) |v(t)| dt = ∫(0 to π/2) |3sint - 3/2| dt

Consider the x-intercept(s) of v(t) = 3sint - 3/2 in [0,π/2]
0 = 3sint - 3/2
sint = 1/2
t = π/6

when t < π/6, v(t) is negative
when t < π/6, v(t) is positive

Thus ∫(0 to π/2) |3sint - 3/2| dt = -∫(0 to π/6) 3sint - 3/2 dt + ∫(π/6 to π/2) 3sint - 3/2 dt = (3cost + 3/2t)|(0 to π/6) + (-3cost - 3/2t)|(π/6 to π/2) = 3cos(π/6) + 3/2(π/6) - 3cos(0) - 3/2(0) - 3cos(π/2) - 3/2(π/2) + 3cos(π/6) + 3/2(π/6) = 2(3√3/2 + π/4) - 3 - 3π/4 = 3√3 + π/2 - 3 - 3π/4 = 1.41

Is that right?

∫3cost dt = 3sint + C

for v(0) = -3/2 = 3sin0 + C
C = -3/2

∫(0 to π/2) 3sint - 3/2 dt = -3cost - 3/2 t |(0 to π/2) = -3cos(π/2) - 3/2(π/2) + 3cos(0) + 3/2(0) = 0.64

And I don't know anymore.

Consider the graph of a(t) = 3 cost


After integrating, consider the graph of v(t) = 3sint-3/2


Now consider the graph of x(t) = -3cost-3/2t

This is really a big spoiler. -__________-

So, what's the answer?

Maybe it is possible to find the distance by calculating the length of the curve.

x(t) = -3cost-3/2t, x'(t) = v(t) = 3sint-3/2

∫(0 to π/2) (1 + (3sint-3/2)^2)^(1/2) dt = ∫(0 to π/2) (1 + 9(sint)^2 - 9sint + 9/4))^(1/2) dt = ∫(0 to π/2) (9(sint)^2 - 9sint + 13/4)^(1/2) dt = function cannot be integrate
Looks like it is impossible to verify be using other method.

Ok, just look if I did any mistake.

I don't want to read computer language.

Then what's your answer?

I didn't do it yet.

Ok, so what next? Are we done?

Have we done the improper integrals?

No, we haven't, but we should.

By the way, do you know the answer for Evil-Mashimaro's question?

I never had this question. I'm not in your class.

Can you do it? Just to verify the answer.

Yes, but no. I don't want to waste time on writing long ugly equations with the key board. I'll ask Spirit for this. XD

There's are questions again? No fear cuz I am back! Ohohoho!

Can you do the Improper Integrals exercise sheet?

Can you verify Evil-Mashimaro's answer?

Wow. One at the time! Let be finish swallowing my medication first.

XD JellyFish, were you waiting for Spirit. Every time he pops out you pops out.

I was online a the time.

Then did you try Evil-Mashimaro's question?

Not really. I got the same answer after viewing her answers. ^^lll

So you want me do all of them. You know that the first ones are pretty easy and it would just be a waste of time for all of us.
We can still go with the 5 last ones.