Find the arc length of the curve described by the parametric equation over the given interval:
x=t^(2) + 1
y=2t - 3
--> 0<t<1
To find the arc length of a curve described by parametric equations, you can use the arc length formula:
s = ∫(sqrt(dx/dt)^2 + (dy/dt)^2) dt
where dx/dt and dy/dt represent the derivatives of x and y with respect to t.
Let's find dx/dt and dy/dt:
dx/dt = 2t
dy/dt = 2
Now we can substitute these values into the arc length formula:
s = ∫(sqrt((2t)^2 + 2^2) dt)
simplifying the expression:
s = ∫(sqrt(4t^2 + 4) dt)
Next, we integrate the expression between the limits of integration: 0 and 1.
s = ∫[0 to 1](sqrt(4t^2 + 4) dt)
Since this integral does not have a simple closed form, we can approximate the integral using numerical methods such as numerical integration or approximating the integral as a Riemann sum. One common numerical method is the trapezoidal rule.
Using the trapezoidal rule with n intervals:
s ≈ Δt * [f(a)/2 + f(a + Δt) + f(a + 2Δt) + ... + f(b - Δt) + f(b)/2]
where Δt = (b - a) / n, a is the lower limit of integration (0 in this case), and b is the upper limit of integration (1 in this case).
Let's approximate the integral using the trapezoidal rule with n = 4 intervals:
Δt = (1 - 0) / 4 = 1/4
s ≈ (1/4) * [f(0)/2 + f(1/4) + f(1/2) + f(3/4) + f(1)/2]
s ≈ (1/4) * [sqrt(4(0)^2 + 4) + sqrt(4(1/4)^2 + 4) + sqrt(4(1/2)^2 + 4) + sqrt(4(3/4)^2 + 4) + sqrt(4(1)^2 + 4)]
Now, you can calculate the individual terms and sum them up to get the approximate arc length.