Calculate the arc length of the curve $$ y = x^{2} $$ from $$ x = 1 $$ to $$ x = 3 $$.

Question

UpStudy AI · Accepted Answer

To calculate the arc length of the curve $$ y = x^{2} $$ from $$ x = 1 $$ to $$ x = 3 $$, we'll use the formula for the arc length of a function $$ y = f(x) $$:

$$
\text{Arc Length} = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
$$

### Step 1: Compute the Derivative

Given $$ y = x^2 $$, the derivative is:

$$
\frac{dy}{dx} = 2x
$$

### Step 2: Set Up the Integral

Plugging the derivative into the arc length formula:

$$
\text{Arc Length} = \int_{1}^{3} \sqrt{1 + (2x)^2} \, dx = \int_{1}^{3} \sqrt{1 + 4x^2} \, dx
$$

### Step 3: Evaluate the Integral

The integral $$ \int \sqrt{4x^2 + 1} \, dx $$ can be solved using a standard integration formula:

$$
\int \sqrt{4x^2 + 1} \, dx = \frac{x}{2} \sqrt{4x^2 + 1} + \frac{1}{4} \ln\left(2x + \sqrt{4x^2 + 1}\right) + C
$$

Evaluating from $$ x = 1 $$ to $$ x = 3 $$:

$$
\begin{align*}
\text{Arc Length} &= \left[ \frac{3}{2} \sqrt{37} + \frac{1}{4} \ln(6 + \sqrt{37}) \right] - \left[ \frac{1}{2} \sqrt{5} + \frac{1}{4} \ln(2 + \sqrt{5}) \right] \
&= \frac{3}{2} \sqrt{37} - \frac{1}{2} \sqrt{5} + \frac{1}{4} \ln\left(\frac{6 + \sqrt{37}}{2 + \sqrt{5}}\right)
\end{align*}
$$

### Step 4: Numerical Approximation

Let's approximate the values:

- $$ \sqrt{37} \approx 6.08276 $$
- $$ \sqrt{5} \approx 2.23607 $$
- $$ \ln(6 + \sqrt{37}) \approx \ln(12.08276) \approx 2.4918 $$
- $$ \ln(2 + \sqrt{5}) \approx \ln(4.23607) \approx 1.4436 $$

Plugging these into the expression:

$$
\begin{align*}
\text{Arc Length} &\approx \frac{3}{2} \times 6.08276 - \frac{1}{2} \times 2.23607 + \frac{1}{4} \times (2.4918 - 1.4436) \
&\approx 9.12414 - 1.11803 + 0.26205 \
&\approx 8.26816
\end{align*}
$$

### Final Answer

The arc length of the curve $$ y = x^2 $$ from $$ x = 1 $$ to $$ x = 3 $$ is approximately **8.27 units**.
The arc length of the curve $$ y = x^2 $$ from $$ x = 1 $$ to $$ x = 3 $$ is approximately **8.27 units**.

Calculate the arc length of the curve from to .

Upstudy AI Solution

Answer

Solution

Step 1: Compute the Derivative

Step 2: Set Up the Integral

Step 3: Evaluate the Integral

Step 4: Numerical Approximation

Final Answer

The Deep Dive

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