Question

Express the arc length s of y=4x² for 0≤x≤4 as an integral using the parametrization r(t)=⟨t⁴,4t⁸⟩ but do not evaluate it.

s(t) = ∫ √³ of ______________
0→4

Answer 1

To express the arc length of y=4x² for 0≤x≤4 using the parametrization r(t)=⟨t⁴,4t⁸⟩, find the derivative of r(t) and use it to calculate the derivative of x with respect to t. Then, use the formula for arc length to construct the integral expression.

Step-by-step explanation:

To express the arc length of y=4x² for 0≤x≤4 using the parametrization r(t)=⟨t⁴,4t⁸⟩, we need to find the derivative of r(t) and use it to calculate the derivative of x with respect to t. Then, we can use the formula for arc length to construct an integral expression.

Step 1: Find the derivative of r(t).

r'(t) = ⟨d/dt(t⁴),d/dt(4t⁸)⟩ = ⟨4t³,32t⁷⟩

Step 2: Calculate the derivative of x with respect to t.

dx/dt = d/dt(t⁴) = 4t³

Step 3: Use the formula for arc length to construct the integral expression.

s(t) = ∫ √(1+(dx/dt)²) dt from 0 to 4