Question

Consider the following. x = t2 − 2t, y = t5, 1 ≤ t ≤ 4 Set up an integral that represents the length of the curve. 4 1 dt Use your calculator to find the length correct to four decimal places.

Answer 1

Answer:

L ≈ 1023.0562

Step-by-step explanation:

We are given;

x = t² - 2t

dx/dt = 2t - 2

Also, y = t^(5)

dy/dt = 5t⁴

The arc length formula is;

L = (α,β)∫√[(dx/dt)² + (dy/dt)²]dt

Where α and β are the boundary points. Thus, applying this to our question, we have;

L = (1,4)∫√[(2t - 2)² + (5t⁴)²]dt

L = (1,4)∫√[4t² - 8t + 4 + 25t^(8)]dt

L = (1,4)∫√[25t^(8) + 4t² - 8t + 4]dt

Using online integral calculator, we have;

L ≈ 1023.0562