Set up, but do not evaluate, an integral for the length of the curve.

Question

x = t + t²

y = t

2 ≤ t ≤ 3

Answer 1

Answer 2

L=∫^b_a √((dx/dt)²+(dy/dt)²) dt

a =2

b=3

dx/dt = 1+2t

dy/dt = 1

You should be able to finish the set up with this information.

Answer 3

To find the length of a curve, it would be as if the function f were a string, and we measure its length with a ruler. Essentially, it's the distance from the start to the end of the ruler, and in terms of the function, it is the distance from the tiny changes in x and y over time, on the interval t∈[2,3]. This is why the formula resembles the distance formula, and what goes in are dx/dt and dy/dt. Setting up the integral, it would be from 2 to 3, integrating √[(dx/dt)²+(dy/dt)²] with respect to t, where (dx/dt)=1+2t and (dy/dt)=1.