evaluate the arc length of the curve 
y
=
e
x
y=e 
x
  from 
x
=
0
x=0 to 
x
=
1
x=1

To find the arc length of the curve \( y = e^x \) from \( x = 0 \) to \( x = 1 \), we use the formula for arc length in calculus:

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

### Step 1: Find the derivative of \( y \)
The curve is given by \( y = e^x \), so:

\[
\frac{dy}{dx} = e^x
\]

### Step 2: Set up the arc length formula
Substitute the derivative into the arc length formula:

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

### Step 3: Evaluate the integral
This integral doesn't have a simple elementary form, but it can be approximated numerically.

### Approximation:
We can approximate this integral using numerical methods. The integral evaluates approximately to:

\[
L \approx 1.919
\]

Therefore, the arc length of the curve \( y = e^x \) from \( x = 0 \) to \( x = 1 \) is approximately \( 1.919 \).

Would you like me to go into more detail on numerical methods, or do you have any specific questions?

---

Here are 5 related questions to expand on this topic:
1. How would you apply this method to find the arc length for other exponential functions?
2. What happens if the limits of integration change, say from \( x = 1 \) to \( x = 2 \)?
3. How do numerical integration methods like the trapezoidal rule or Simpson’s rule work for such problems?
4. Can we find an exact solution for some functions using this method?
5. How does the arc length formula change for parametric curves?

**Tip:** In arc length problems, simplifying the integrand as much as possible before solving often makes numerical or exact methods easier.

Evaluate the arc length of the curve y = e^x from x = 0 to x = 1.

Learn how to evaluate the arc length of the curve y = e^x from x = 0 to x = 1. This problem involves key calculus concepts, including the arc length formula, exponential function derivatives, and numerical integration methods. Suitable for advanced high school and college-level students.

Math Problem Statement

evaluate the arc length of the curve y

e x y=e x from x

0 x=0 to x

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