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Find the arc length of the curve y=1/3(x²+2)³/² on the interval [0,3]. What is the correct arc length for this curve?

User JZweige
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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].

User Cazzer
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