To find how far the particle travels in the first 6 seconds, we need to calculate the total displacement, which is given by the integral of the velocity function $v(t) = t^2 e^{-t}$ over the interval from $t = 0$ to $t = 6$.

This distance is represented by the definite integral:

$\text{Distance} = \int_0^6 v(t) \, dt = \int_0^6 t^2 e^{-t} \, dt$

We will now solve this integral using integration by parts.

Step 1: Integration by Parts

The formula for integration by parts is:

$\int u \, dv = uv - \int v \, du$

Let:

• $u = t^2$ (which simplifies upon differentiation)
• $dv = e^{-t} dt$ (which is easy to integrate)

Now, differentiate and integrate:

• $du = 2t \, dt$
• $v = -e^{-t}$

Step 2: Apply Integration by Parts

Now apply the integration by parts formula:

$\int t^2 e^{-t} \, dt = -t^2 e^{-t} + \int 2t e^{-t} \, dt$

We need to integrate $2t e^{-t}$. We can use integration by parts again for this.

Let:

• $u = 2t$
• $dv = e^{-t} dt$

Then:

• $du = 2 \, dt$
• $v = -e^{-t}$

So,

$\int 2t e^{-t} \, dt = -2t e^{-t} + \int 2 e^{-t} \, dt$

Now integrate $2 e^{-t}$:

$\int 2 e^{-t} \, dt = -2e^{-t}$

Thus, the integral of $2t e^{-t}$ becomes:

$\int 2t e^{-t} \, dt = -2t e^{-t} - 2e^{-t}$

Step 3: Combine Results

Now combine everything back into the original expression:

$\int t^2 e^{-t} \, dt = -t^2 e^{-t} + (-2t e^{-t} - 2e^{-t})$

Simplifying:

$\int t^2 e^{-t} \, dt = -(t^2 + 2t + 2) e^{-t}$

Step 4: Evaluate the Definite Integral

Now, evaluate the definite integral from $t = 0$ to $t = 6$:

$\int_0^6 t^2 e^{-t} \, dt = \left[ -(t^2 + 2t + 2) e^{-t} \right]_0^6$

First, evaluate at $t = 6$:

$-(6^2 + 2(6) + 2) e^{-6} = -(36 + 12 + 2) e^{-6} = -50e^{-6}$

Next, evaluate at $t = 0$:

$-(0^2 + 2(0) + 2) e^{-0} = -2$

Step 5: Final Answer

Now subtract the two values:

$\int_0^6 t^2 e^{-t} \, dt = -50e^{-6} - (-2) = -50e^{-6} + 2$

Thus, the exact distance the particle travels in the first 6 seconds is:

$2 - 50e^{-6} \, \text{feet}$

Would you like further clarification on any step, or do you have any specific questions?

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Here are 5 related questions for you:

1. How does integration by parts work in general?
2. How would the result change if the velocity function were different?
3. What does the value $e^{-6}$ approximately represent in decimal form?
4. Can you explain the physical meaning of the velocity function $v(t)$?
5. How would you calculate the particle's acceleration?

Tip: When solving problems with exponential functions, always check if you can simplify using known integrals or rules like integration by parts!