Vector Calculus line integral help

Hello everyone, I am reviewing for my vector calculus exam and one of the line integral problems my professor recommended us to look over isn't like any of the other problems I have done, so could someone please tell me how to solve this. (The answer to this problem is 2.8)

Here is the problem:

Compute the line integral of the scalar function over the curve:

f(x, y) = sqrt(1+9xy) y = x^3 for 0<=x<=1


I know that the formula for this type of question is:

ɾ
| f(c(t)) ||c'(t)|| dt
ᴊ C

but I'm not sure how to solve this if I don't know c(t), so could someone please help me out with this because I'm very confused. (I only need to know how to find c(t))

Thanks,

Kevin

To find c(t), you will need to parametrize the curve.
Since y = x^3 for 0 <= x <= 1, we can use x = t, y = t^3, 0 <= t <= 1.
Thus we can use c(t) = for 0 <= t <= 1.

So the line integral of f(x, y) = sqrt(1+9xy) along the curve is

integral 0 to 1 of f(t, t^3) ||d/dt of || dt
= integral 0 to 1 of sqrt(1+9t*t^3) ||<1, 3t^2>|| dt
= integral 0 to 1 of sqrt(1+9t^4) * sqrt(1^2+(3t^2)^2) dt
= integral 0 to 1 of sqrt(1+9t^4) * sqrt(1+9t^4) dt
= integral 0 to 1 of (1+9t^4) dt
= (t + (9/5)t^5) evaluated from t = 0 to t = 1
= (1 + (9/5)(1^5)) - (0 + (9/5)(0^5))
= 14/5
= 2.8

Lord bless you today!