Estimate the length of the curve f(x)= x^2+1 on -2 ≤ x ≤ 2. Using n=4

To estimate the length of the curve \( f(x) = x^2 + 1 \) over the interval \( -2 \leq x \leq 2 \) using \( n = 4 \) subintervals, we use the formula for the arc length of a function, which is given by:

\[
L \approx \sum_{i=1}^{n} \sqrt{1 + \left( \frac{dy}{dx} \bigg|_{x_i^*} \right)^2} \Delta x
\]

### Step 1: Determine the derivative of \( f(x) \)
The first derivative of \( f(x) = x^2 + 1 \) is:
\[
\frac{dy}{dx} = 2x
\]

### Step 2: Set up the interval and subintervals
Given \( -2 \leq x \leq 2 \), the interval length is \( 4 \). With \( n = 4 \) subintervals, the width of each subinterval is:
\[
\Delta x = \frac{4}{4} = 1
\]
The subintervals are:
\[
x_0 = -2, \quad x_1 = -1, \quad x_2 = 0, \quad x_3 = 1, \quad x_4 = 2
\]

### Step 3: Calculate the function's derivative at the midpoint of each subinterval
We use the midpoints for better accuracy. The midpoints \( x_i^* \) are:
\[
x_0^* = -1.5, \quad x_1^* = -0.5, \quad x_2^* = 0.5, \quad x_3^* = 1.5
\]
Calculate \( \frac{dy}{dx} \) at these points:
\[
\frac{dy}{dx} \bigg|_{x_0^*} = 2(-1.5) = -3, \quad \frac{dy}{dx} \bigg|_{x_1^*} = 2(-0.5) = -1
\]
\[
\frac{dy}{dx} \bigg|_{x_2^*} = 2(0.5) = 1, \quad \frac{dy}{dx} \bigg|_{x_3^*} = 2(1.5) = 3
\]

### Step 4: Calculate the length of the curve
Now, plug these into the arc length formula:
\[
L \approx \sum_{i=1}^{4} \sqrt{1 + \left( \frac{dy}{dx} \bigg|_{x_i^*} \right)^2} \Delta x
\]
\[
L \approx \sqrt{1 + (-3)^2} \cdot 1 + \sqrt{1 + (-1)^2} \cdot 1 + \sqrt{1 + 1^2} \cdot 1 + \sqrt{1 + 3^2} \cdot 1
\]
\[
L \approx \sqrt{1 + 9} + \sqrt{1 + 1} + \sqrt{1 + 1} + \sqrt{1 + 9}
\]
\[
L \approx \sqrt{10} + \sqrt{2} + \sqrt{2} + \sqrt{10}
\]
\[
L \approx 3.162 + 1.414 + 1.414 + 3.162
\]
\[
L \approx 9.152
\]

So, the estimated length of the curve \( f(x) = x^2 + 1 \) over the interval \( -2 \leq x \leq 2 \) is approximately **9.152** units.

Would you like any details or have any questions?

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Here are 5 related questions:

1. How would the estimated curve length change if we increased the number of subintervals \( n \) to 8?
2. How do we find the exact length of the curve instead of estimating it?
3. Can this method be used for any function \( f(x) \)? If not, what are the limitations?
4. What happens to the estimated length if we use unequal subintervals instead of equal ones?
5. How can you estimate the length of a curve in three dimensions?

**Tip:** For more accurate curve length estimates, increase the number of subintervals \( n \), especially when the function's derivative changes rapidly.

Estimate the length of the curve f(x)= x^2+1 on -2 ≤ x ≤ 2. Using n=4.

Learn how to estimate the length of the curve f(x) = x^2 + 1 over the interval [-2, 2] using 4 subintervals. This problem involves key calculus concepts such as derivatives and the arc length formula, making it ideal for college-level students studying calculus.

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