Arc length of the curve

Question

Find the arc length of the curve on the interval.

r(t) =

t2 + 1, 8t3 + 7

, −1 ≤ t ≤ 0

Answer 1

Hi Lew,

I'm not going to work out the problem for you (that wouldn't help you learn).

I will give you some advice however. Note: Bold means a vector quantity (an arrow).

You are given a set of parametric equations <x(t) , y(t)> and need the arc length.

If you think in terms of physics <x'(t) , y'(t)> gives the velocity in each direction.

x'(t) is the horiztonal velocity and y'(t) is the vertical velocity.

These are two vectors so you add them tail to tip.

The speed is the length of the resultant vector v(t) is denoted v(t).

Hint: Use the Pythagorean Theorem to find v(t).

So finally, how do we find the distance traveled?

For a constant speed the answer is trival d = vt .

In this case the speed is changing, so we must look at small time intervals delta t and then sum.

d = ∑ v Δt

throught the magic of the limit this turns into ∫ v dt

Notice that the distance traveled depends on speed only, not the direction so v(t) is used.

In other words if I travel at 10 m/s to the right for 2 seconds and then 6 m/s to the left for 2 seconds the distance traveled is (10)(2) + (6)(2) = 32 m. We use only the length of the velocity vector not the direction for distance.

That's the explanation. See if you can find v(t) = the speed (which is the length of v(t) ).

Remember, to find v(t) draw a triangle with x'(t) and y'(t) then find the hypotenuse.

After that you can set up your integral.

Please let me know if you get stuck (just reply on here). I can check your work.