Find the arc length of the curve y = ex for x ∈ [0, 1/2].

To find the arc length of the curve \( y = e^x \) over the interval \( x \in [0, \frac{1}{2}] \), we can use the arc length formula for a function \( y = f(x) \):

\[
L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]

### Step 1: Find the derivative \( \frac{dy}{dx} \)
For \( y = e^x \), the derivative is:

\[
\frac{dy}{dx} = \frac{d}{dx} \left( e^x \right) = e^x
\]

### Step 2: Set up the arc length formula
Now substitute \( \frac{dy}{dx} = e^x \) into the arc length formula:

\[
L = \int_0^{\frac{1}{2}} \sqrt{1 + \left( e^x \right)^2} \, dx
\]

This simplifies to:

\[
L = \int_0^{\frac{1}{2}} \sqrt{1 + e^{2x}} \, dx
\]

### Step 3: Evaluate the integral
This integral does not have a simple elementary antiderivative, so it is typically evaluated using numerical methods. I can compute it numerically for you.

Let's proceed with that.The arc length of the curve \( y = e^x \) over the interval \( x \in [0, \frac{1}{2}] \) is approximately 0.821.

Do you want any further details or have questions?

Here are 5 related questions to expand on this concept:
1. How would the arc length change if the interval were extended to \( [0, 1] \)?
2. How do you calculate arc lengths for parametric equations?
3. What is the geometric interpretation of arc length in terms of a curve's shape?
4. How does changing the curve to \( y = e^{2x} \) affect the arc length calculation?
5. What are common methods for approximating integrals that cannot be solved analytically?

**Tip**: When evaluating integrals numerically, the method of choice (like Simpson's rule or trapezoidal rule) can affect the precision and speed of the calculation.

Find the arc length of the curve y = e^x for x ∈ [0, 1/2].

Calculate the arc length of the curve y = e^x over the interval [0, 1/2] using calculus. This solution covers key concepts in arc length calculation, exponential function derivatives, and numerical integration methods. Suitable for students in Grades 11-12.

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