Question

Find the exact length of the portion of the curve shown in blue.

Answer 1

The exact length of the portion of the curve shown in blue cannot be determined without specific mathematical information or an equation for the curve.

Step-by-step explanation:

To find the exact length of a curve, we typically use calculus and integrate the curve's derivative over a given interval. However, without additional details about the mathematical form of the curve or the specific interval in question, it's impossible to provide an exact length.

If the curve is defined by a function y = f(x), the arc length (s) is given by the integral:

`\[ s = \int_(a)^(b) \sqrt{1 + \left((dy)/(dx)\right)^2} \,dx \]`

where a and b are the limits of integration. Without these specifics or the equation of the curve, we cannot proceed with the calculation. Additionally, if the curve is not defined mathematically, determining its exact length becomes an undefined problem.

In summary, without a specific mathematical representation of the curve or the interval over which its length is to be calculated, it is not possible to provide an exact length. The process involves mathematical tools and information that are not provided in the given question.

Answer 2

The integra is ∫ √((6 + 6sin(θ))² + 36cos²(θ)) dθ

Step-by-step explanation:

To find the exact length of the portion of the curve, we need to calculate the arc length. The formula for the arc length is given by:

As = ∫ √(r² + (dr/dθ)²) dθ

In this case, the curve is represented by the equation r = 6 + 6sin(θ). To calculate the length, we need to substitute this equation into the formula and integrate.

After simplification, the integral becomes: ∫ √(6 + 6sin(θ)² + 36cos²(θ)) dθ

By evaluating this integral, we can find the exact length of the portion of the curve.

Your question is incomplete, but most probably the full question was:

Find the exact length of the portion of the curve shown in blue.

r=6+6sin(θ)