Find the line integral of 𝑓(𝑥, 𝑦) = 𝑦𝑒 # !along the curve r(t) = 4t i – 3t j, -1 ≤ t ≤ 2

To solve this problem, we need to follow these steps:

1. First, we need to parameterize the function f(x, y) in terms of t using the given curve r(t) = 4t i – 3t j. From r(t), we can see that x = 4t and y = -3t. So, f(x, y) becomes f(t) = -3t * e.

2. Next, we need to find the derivative of the vector function r(t). The derivative dr/dt = 4 i - 3 j.

3. Then, we need to find the magnitude of the derivative, which is ||dr/dt|| = sqrt((4)^2 + (-3)^2) = 5.

4. Now, we can compute the line integral from t = -1 to t = 2. The line integral ∫C f(r(t)) ||dr/dt|| dt becomes ∫ from -1 to 2 of (-3t * e) * 5 dt.

5. Finally, we need to solve this integral. The integral of -15t * e dt from -1 to 2 is [-15/2 * e * t^2] from -1 to 2 = -15/2 * e * 4 - (-15/2 * e * 1) = -30e + 7.5e = -22.5e.

So, the line integral of f(x, y) = y * e along the curve r(t) = 4t i – 3t j from t = -1 to t = 2 is -22.5e.