https://file.aiquickdraw.com/mathbot/file/1729134633536u5pa57ch.jpg

The problem asks to find the surface area generated by rotating the given curve about the **y-axis**, with the parametric equations:

\[
x = e^t - t, \quad y = 4e^{t/2}, \quad 0 \leq t \leq 4
\]

To find the surface area generated by rotating a parametric curve \(x(t)\) and \(y(t)\) around the y-axis, the formula is:

\[
S = 2\pi \int_{t_1}^{t_2} x(t) \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt
\]

### Step-by-step solution:

1. **Differentiate \(x(t)\) and \(y(t)\)**:
   - \( x(t) = e^t - t \)
     \[
     \frac{dx}{dt} = e^t - 1
     \]
   - \( y(t) = 4e^{t/2} \)
     \[
     \frac{dy}{dt} = 2e^{t/2}
     \]

2. **Set up the surface area integral**:
   - We substitute the expressions for \(x(t)\), \(\frac{dx}{dt}\), and \(\frac{dy}{dt}\) into the surface area formula:
     \[
     S = 2\pi \int_0^4 (e^t - t) \sqrt{(e^t - 1)^2 + (2e^{t/2})^2} \, dt
     \]
     Simplifying the expression under the square root:
     \[
     S = 2\pi \int_0^4 (e^t - t) \sqrt{(e^t - 1)^2 + 4e^t} \, dt
     \]

Now, the integral can be computed either analytically or numerically. However, evaluating this directly by hand can be quite complex, so typically, a numerical method would be used (or a tool like a calculator or software for integral evaluation).

Would you like assistance with calculating this integral, or would you prefer further details on any specific part of the solution?

### Relative Questions:
1. What are parametric equations and how are they used in finding surface areas?
2. How do you differentiate parametric equations like \(x(t)\) and \(y(t)\)?
3. Why is the square root of the sum of squares used in the surface area formula?
4. How does rotating around the y-axis differ from rotating around the x-axis?
5. What numerical methods could be used to evaluate this integral?

### Tip:
When calculating surface areas of revolved parametric curves, be careful with setting up the correct limits of integration and expressions for derivatives to avoid mistakes in more complex problems.

Find the surface area generated by rotating the given curve about the y-axis: x = e^t - t, y = 4e^{t/2}, 0 ≤ t ≤ 4.

Calculate the surface area generated by rotating the parametric curve x = e^t - t, y = 4e^{t/2}, 0 ≤ t ≤ 4 about the y-axis. This problem involves parametric equations and surface area calculation using integration. Suitable for calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation