Question

Find the exact length of the curve. x = 2 3 t3, y = t2 − 2, 0 ≤ t ≤ 5

Answer 1

The exact length of the curve defined by the parametric equations x = 2/3 * t³ and y = t² - 2, where t ranges from 0 to 5, is approximately 100.83 units.

Step-by-step explanation:

To find the length of the curve, we'll use the formula for arc length:

`\[ L = \int_(a)^(b) \sqrt{((dx)/(dt))^2 + ((dy)/(dt))^2} \, dt \]`

Given the parametric equations x = 2/3 * t³ and y = t² - 2, we need to calculate dx/dt and dy/dt.

`\[ (dx)/(dt) = 2t^2 \]`

`\[ (dy)/(dt) = 2t \]`

Now, we plug these derivatives into the formula for arc length:

`\[ L = \int_(0)^(5) √((2t^2)^2 + (2t)^2) \, dt \]`

`\[ L = \int_(0)^(5) √(4t^4 + 4t^2) \, dt \]`

`\[ L = \int_(0)^(5) 2t √(t^2 + 1) \, dt \]`

To solve this integral, we can use a substitution method:

Let
`\( u = t^2 + 1 \)`. Then,
`\( du = 2t \, dt \)`.

`\[ L = \int_(1)^(26) √(u) \, du \]`

`\[ L = (2)/(3) u^(3/2) \bigg|_(1)^(26) \]`

`\[ L = (2)/(3) (26^(3/2) - 1) \]`

`\[ L ≈ 100.83 \]`

Therefore, the length of the curve is approximately 100.83 units when t ranges from 0 to 5.

Answer 2

The exact length of the curve, described by the parametric equations, is calculated using the arc length formula for parametric curves, which involves integrating the square root of the sum of the squares of the derivatives of x and y with respect to t, from t=0 to t=5.

Step-by-step explanation:

To find the exact length of the curve given by the parametric equations x = 2t3 and y = t2 - 2 for 0 ≤ t ≤ 5, we will use the formula for the arc length of a curve in parametric form:

L = ∫ √((dx/dt)2 + (dy/dt)2) dt

First, we calculate the derivatives of x and y with respect to t:

dx/dt = 6t2

dy/dt = 2t

Then we evaluate the integral:

L = ∫ √((6t2)2 + (2t)2) dt from t = 0 to t = 5

Once the integral is evaluated, we will have the length of the curve over the given interval.