Calculate the length of the path over the given interval: c(t) = (3t^2, 4t^3), 2<t<5
To calculate the length of the path over the given interval, we need to use the arc length formula. The formula for the length of a path in two dimensions is:
L = ∫[a, b] √(dx/dt)^2 + (dy/dt)^2 dt
In this case, we have c(t) = (3t^2, 4t^3). To calculate the length, we need to find the derivativess
dx/dt = d/dt (3t^2) = 6t
dy/dt = d/dt (4t^3) = 12t^2
Now, we can substitute these derivatives back into the arc length formula:
L = ∫[2, 5] √(6t)^2 + (12t^2)^2 dt
L = ∫[2, 5] √36t^2 + 144t^4 dt
L = ∫[2, 5] √(36t^2 + 144t^4) dt
Since this integral is not easily solved by hand, we can use computational methods such as numerical integration to approximate the value. There are several numerical integration techniques available, such as the trapezoidal rule or Simpson's rule, which can be implemented in software programs or calculators to compute the approximate value of the integral.
Once we have the approximate value of the integral, we will have the length of the path over the given interval.
s = ∫[2,5] √(dx^2+dy^2) dt
= ∫[2,5] √((6t)^2 + (12t^2)^2) dt
= ∫[2,5] √(36t^2 + 144t^4) dt
= ∫[2,5] 6t√(1+4t^2) dt
Letting u = 1+4t^2, we have
= ∫[17,101] 3/4 √u du
= (3/4)(2/3) u^(3/2) [17,101]
= 1/2 (101√101 - 17√17)