Find the exact length of the curve defined by x=13y⁢(y−3) for 9≤y≤16.

The length is ∫ 9 16 1 + [ 2 y − 3 3 ] 2 d y

Find the exact length of the curve defined by x = 1 3 y ⁢ ( y − 3 ) for 9 ≤ y ≤ 16 .

The length is ∫ 9 16 1 + [ 2 y − 3 3 ] 2 d y

The length is ∫ 9 16 1 + [ y − 3 3 ] 2 d y

The length is ∫ 9 16 1 + [ 2 y + 3 3 ] 2 d y

The length is ∫ 9 16 1 + [ y + 3 3 ] 2 d y

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Introduction to Functions

Calculus

0. Functions

Introduction to Functions

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Multiple Choice

Find the exact length of the curve defined by $x = \frac{1}{3} y (y - 3)$ for $9 \leq y \leq 16$ .

The length is

\int_{9}^{16} \sqrt{1 + {[\frac{y - 3}{3}]}^{2}} d y

The length is

\int_{9}^{16} \sqrt{1 + {[\frac{2 y - 3}{3}]}^{2}} d y

The length is

\int_{9}^{16} \sqrt{1 + {[\frac{2 y + 3}{3}]}^{2}} d y

The length is

\int_{9}^{16} \sqrt{1 + {[\frac{y + 3}{3}]}^{2}} d y

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Verified step by step guidance

Step 1: Recall the formula for the arc length of a curve defined parametrically or explicitly. For a curve defined as x = f(y), the arc length is given by ∫_{a}^{b} sqrt{1 + (dx/dy)^2} dy, where dx/dy is the derivative of x with respect to y.

Step 2: Compute dx/dy for the given function x = (1/3)y(y - 3). Use the product rule to differentiate x with respect to y. The product rule states that if x = u*v, then dx/dy = u'(v) + u(v').

Step 3: Apply the product rule to x = (1/3)y(y - 3). Here, u = (1/3)y and v = (y - 3). Differentiate u and v separately: u' = (1/3) and v' = 1. Substitute these into the product rule to find dx/dy.

Step 4: Simplify dx/dy to obtain the derivative. Substitute dx/dy into the arc length formula ∫_{9}^{16} sqrt{1 + (dx/dy)^2} dy.

Step 5: Simplify the integrand sqrt{1 + (dx/dy)^2} to match one of the given options. Verify which option corresponds to the correct simplification of the integrand and set up the integral for evaluation.