Find the arc length of the graph of the function $y=\frac{2}{3}{x}^{\frac{3}{2}}+\frac{1}{3}$ over the interval $0\le x\le 9$.

• A. $12$
• B. $18$
• C. $9$
• D. $15$

Answer: (A)

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]. The formula is given by: L = ∫[a, b] √(1 + (dy/dx)^2) dx.
2. Step 2: Differentiate the given function y = (2/3)x^(3/2) + (1/3) with respect to x to find dy/dx. Using the power rule, dy/dx = (2/3) * (3/2)x^(1/2) = x^(1/2).
3. Step 3: Substitute dy/dx = x^(1/2) into the arc length formula. This gives: L = ∫[0, 9] √(1 + (x^(1/2))^2) dx = ∫[0, 9] √(1 + x) dx.
4. Step 4: Evaluate the integral ∫[0, 9] √(1 + x) dx. To do this, use a substitution method. Let u = 1 + x, so du = dx. Adjust the limits of integration: when x = 0, u = 1; when x = 9, u = 10. The integral becomes ∫[1, 10] √u du.
5. Step 5: Solve the integral ∫[1, 10] √u du. Rewrite √u as u^(1/2) and use the power rule for integration: ∫u^(1/2) du = (2/3)u^(3/2) + C. Apply the limits of integration to find the arc length.