Find the exact length of the curve $y=2\surd 3x^{\frac{3}{2}}$ for $0\le x\le 6$.

• A. $56$
• B. $36+12\surd 19$
• C. $74+12\surd 37$
• D. $74$

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the formula for the arc length of a curve y = f(x) over the interval [a, b]: \( L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \). This formula calculates the exact length of the curve by integrating the square root of 1 plus the square of the derivative of the function.
2. Step 2: Compute the derivative of the given function \( y = 2 \sqrt{3} x^{3/2} \). Using the power rule, \( \frac{dy}{dx} = 2 \sqrt{3} \cdot \frac{3}{2} x^{1/2} = 3 \sqrt{3} x^{1/2} \).
3. Step 3: Substitute \( \frac{dy}{dx} \) into the arc length formula. The integrand becomes \( \sqrt{1 + \left( 3 \sqrt{3} x^{1/2} \right)^2} = \sqrt{1 + 27x} \). The arc length formula now reads \( L = \int_0^6 \sqrt{1 + 27x} \, dx \).
4. Step 4: Solve the integral \( \int_0^6 \sqrt{1 + 27x} \, dx \). Use a substitution method: let \( u = 1 + 27x \), so \( du = 27 \, dx \). Adjust the limits of integration accordingly: when \( x = 0 \), \( u = 1 \); when \( x = 6 \), \( u = 163 \). The integral becomes \( \frac{1}{27} \int_1^{163} \sqrt{u} \, du \).
5. Step 5: Evaluate \( \frac{1}{27} \int_1^{163} \sqrt{u} \, du \). The antiderivative of \( \sqrt{u} \) is \( \frac{2}{3} u^{3/2} \). Apply the limits of integration: \( \frac{1}{27} \left[ \frac{2}{3} u^{3/2} \right]_1^{163} \). Simplify the expression to find the exact length of the curve.