Question:

Find the length of the curve defined by the equation $y=x^3-2x^2+3x$ from $x=0$ to $x=4$.

Answer:

To find the length of a curve, we use the formula:

where $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$.

First, let's find $\frac{dy}{dx}$:

Now, let's plug it into the formula for length:

Expanding the expression inside the square root:

Simplifying the expression under the square root:

To evaluate this integral, we can either use numerical methods or use a computer software. Evaluating the integral:

Therefore, the length of the curve defined by the equation $y=x^3-2x^2+3x$ from $x=0$ to $x=4$ is approximately 16.051 units.