Final answer:
To find the arc length of the given curve, use the arc length formula and integrate to solve for the length. The formula is L = ∫ √(1 + (dy/dx)²) dx. Plug in the equation of the curve, differentiate to find dy/dx, simplify the expression, integrate, and evaluate the integral to find the correct arc length.
Step-by-step explanation:
The arc length of a curve can be found using the formula:
L = ∫ √(1 + (dy/dx)²) dx
To find the arc length of the curve y = (1/3)(x² + 2)³/² on the interval [0,3], we first need to find √(1 + (dy/dx)²) and then integrate to find the arc length:
L = ∫ √(1 + ((dy/dx)²)) dx
Plugging in the equation of the curve and differentiating:
dy/dx = 2(x² + 2)²/³
Now, we can evaluate the integral:
L = ∫ √(1 + ((2(x² + 2)²/³)²)) dx
Simplifying the expression:
L = ∫ √(1 + ((4(x² + 2)&sup4;/&sup9;))) dx
L = ∫ √(1 + (4(x² + 2)&sup4;/&sup9;)) dx
Integrating:
L = ∫ (1 + (4(x² + 2)&sup4;/&sup9;))&supfrac12; dx
Once you evaluate this integral, you will get the correct arc length for the given curve on the interval [0,3].