Find the arc length of y = 4x/2 on 1 < x < 2.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Calculus II

### Problem:
Find the arc length of \( y = 4x^{3/2} \) on \( 1 \leq x \leq 2 \).

### Explanation:
This problem requires finding the arc length of the function over the given interval. The arc length \( L \) of a curve defined by \( y = f(x) \) from \( x = a \) to \( x = b \) can be found using the following formula:

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

For the function \( y = 4x^{3/2} \), the first derivative with respect to \( x \):

\[ \frac{dy}{dx} = 4 \cdot \frac{3}{2} x^{1/2} = 6x^{1/2} \]

Now, we substitute \( \frac{dy}{dx} \) into the arc length formula:

\[ L = \int_{1}^{2} \sqrt{1 + (6x^{1/2})^2} \, dx \]
\[ L = \int_{1}^{2} \sqrt{1 + 36x} \, dx \]

This integral can be solved to find the arc length of the curve from \( x = 1 \) to \( x = 2 \).
Transcribed Image Text:### Problem: Find the arc length of \( y = 4x^{3/2} \) on \( 1 \leq x \leq 2 \). ### Explanation: This problem requires finding the arc length of the function over the given interval. The arc length \( L \) of a curve defined by \( y = f(x) \) from \( x = a \) to \( x = b \) can be found using the following formula: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] For the function \( y = 4x^{3/2} \), the first derivative with respect to \( x \): \[ \frac{dy}{dx} = 4 \cdot \frac{3}{2} x^{1/2} = 6x^{1/2} \] Now, we substitute \( \frac{dy}{dx} \) into the arc length formula: \[ L = \int_{1}^{2} \sqrt{1 + (6x^{1/2})^2} \, dx \] \[ L = \int_{1}^{2} \sqrt{1 + 36x} \, dx \] This integral can be solved to find the arc length of the curve from \( x = 1 \) to \( x = 2 \).
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