Question

Find the length of the following​ two-dimensional curve.

r(t)= < t²/2, (2t+1)³/²/3> for 0 ≤ t ≤ 10

Answer 1

To find the length of the curve r(t) = < t²/2, (2t+1)³/²/3> for 0 ≤ t ≤ 10, you need to calculate its arc length using the formula for parametric curves after finding the derivatives of the x and y components. The integrand constructed from the derivatives is complex, and numerical integration techniques might be necessary to find a precise answer.

Step-by-step explanation:

To find the length of the two-dimensional curve r(t) = < t²/2, (2t+1)³/²/3> for 0 ≤ t ≤ 10, we can use the arc length formula for a parametric curve. The formula for the arc length of a curve given by r(t) from a to b is L = ∫_a^b √(−x'(t)−^2 + −y'(t)−^2) dt. Firstly, we need to calculate the derivatives of the x-component, x'(t), and the y-component, y'(t), of r(t).

For the x-component:

• 
  
• x(t) = t²/2
• 
  
• x'(t) = t

For the y-component:

• 
  
• y(t) = (2t+1)³/²/3
• 
  
• y'(t) = (3/2)(2t + 1)²/² ∙ 2

Then, the integrand can be constructed:

• 
  
• √(t² + (3/2)(2t + 1)²/² ∙ 2²)

Next, we would integrate this expression from t = 0 to t = 10.

However, this integral may be very complex and possibly requires numerical integration techniques rather than a simple analytical expression. Thus, in practice, we might need to use computational tools or approximation methods to evaluate it. For homework and tests, such integral is often approximated or certain assumptions are made to simplify the problem.