✅ Perform

📋 Quiz Question

☁️ Updraft Velocity Integration

Alright, let's talk about clouds and calculus. Imagine a specific type of cloud whose updraft velocity, $v(t)$, in meters per second, at time $t$ in seconds, is modeled by the function:

$$v(t) = t \cdot e^{-t/10}$$

To figure out the total displacement of air parcels within this cloud over a period, we'd need to integrate this velocity function. Using the technique of integration by parts, can you find the indefinite integral of this expression?

$$ \int t e^{-t/10} \, dt $$

Which of the following is the correct result?

A. $10e^{-t/10}(t + 10) + C$

B. $-10e^{-t/10}(t - 10) + C$

C. $e^{-t/10}(t + 10) + C$

D. $-10e^{-t/10}(t + 10) + C$

Take your time, and let me know what you come up with. If you get stuck, I can offer a hint.

• Difficulty: Medium
• Topic: Calculus (Integration by Parts)

🔍 Doublecheck Work

🔬 Check

You've selected option D for the indefinite integral of $v(t) = t \cdot e^{-t/10}$.

📋 Feedback

That's correct! Option D is the right answer. 🥳

You navigated the integration by parts like a seasoned cloud observer. Here's a quick recap of how we got there:

1. We identified $u = t$ and $dv = e^{-t/10} \, dt$.
2. From this, we found $du = dt$ and $v = -10e^{-t/10}$.
3. Applying the integration by parts formula, $\int u \, dv = uv - \int v \, du$, we got $t(-10e^{-t/10}) - \int (-10e^{-t/10}) \, dt$.
4. Simplifying and integrating the remaining term led us to the final answer: $-10e^{-t/10}(t + 10) + C$.

Nicely done! Ready for another cloud-themed calculus challenge?