How do I find the length of curve f(x)=3x^2-2x+3 on the interval [1,4]? I have no idea on how to start this. Anyways, thanks in advance!

The length of a curve for the interval [x1,x2] is given by the integral:

x2
∫sqrt(1+(dy/dx)^2)dx
x1

and where dy/dx = 6x-2

For the present case, as an estimate,
y(1)=4, y(4)=43, so you'd expect the length to be a little over 39, or more precisely, 39.1414... approximately.

Post if you need more help.

length of curve

= ∫√(1 + (dy/dx)^2 ) dx from a to b

so for yours
length = ∫(1 + (6x-2)^2 )^(1/2) dx from 1 to 4
= ∫(36x^2 -24x + 5)^(1/2) dx

Now the mess begins.
At this point I will admit that my integration skills are so rusty, that I can't see a way out.

If it's any help, the substitution 6x-2 = sinh(u) will simplify things a bit.

To find the length of a curve on a given interval, we need to use a formula called the arc length formula. In this case, the arc length formula for a function f(x) on the interval [a, b] is given by:

L = ∫[a,b] √(1 + (f'(x))^2) dx

In this formula, f'(x) represents the derivative of the function f(x) and dx represents the differential element in the integral.

Now, let's find the derivative of the function f(x):
f(x) = 3x^2 - 2x + 3

Taking the derivative, we get:
f'(x) = 6x - 2

Next, we substitute f'(x) into the arc length formula and evaluate the integral:

L = ∫[1,4] √(1 + (6x - 2)^2) dx

To simplify the integral, we can expand and square the term (6x - 2)^2:
(6x - 2)^2 = 36x^2 - 24x + 4

Now, our integral becomes:
L = ∫[1,4] √(1 + 36x^2 - 24x + 4) dx

Combining like terms, we have:
L = ∫[1,4] √(36x^2 - 24x + 5) dx

To evaluate this integral, you can either use integration techniques (such as substitution or integration by parts) or use numerical methods (such as numerical integration or approximation).

If you choose to use numerical methods, you can use software like Mathematica, Wolfram Alpha, or an online integral calculator to find the numerical value of the integral.

Once you have evaluated the integral, you'll obtain the length L of the curve f(x) on the interval [1, 4].