Calculate the length of the path over the given interval: c(t) = (3t^2, 4t^3), 2<t<5
To calculate the length of a path, we can use the arc length formula. The arc length formula for a curve in two-dimensional space is given by:
L = ∫[a,b] √(1 + (dy/dx)^2) dx
Where L denotes the length of the curve, [a, b] represents the interval over which the curve is defined, and dy/dx represents the derivative of y with respect to x.
In this case, we are given the parametric equations of the curve c(t) = (3t^2, 4t^3) and the interval 2 < t < 5. To find the length of the curve, we need to express it in terms of x and y:
x = 3t^2
y = 4t^3.
Differentiating these equations with respect to t, we obtain:
dx/dt = 6t
dy/dt = 12t^2.
To calculate dy/dx, we divide dy/dt by dx/dt:
dy/dx = (12t^2)/(6t) = 2t.
Now, we plug this into the arc length formula and find the integral:
L = ∫[2,5] √(1 + (2t)^2) dt.
Simplifying the integrand:
L = ∫[2,5] √(1 + 4t^2) dt.
The integral can be solved by using standard integration techniques or numerical methods. Once the integral is evaluated, you will have the length of the path over the given interval (2<t<5).