AP Calculus AB Exam Question:
The curve defined by the equation y = 3x^2 - 2x + 5 is given in the Cartesian plane.
a) Determine the length of the curve over the interval [0, 2] using the concept of definite integrals.
b) Determine the length of the curve over the interval [0, 2] using the distance formula and calculus concepts.
Answer:
a) To determine the length of the curve over the interval [0, 2] using definite integrals, we need to calculate the integral of the square root of the sum of the squares of the derivatives of x and y with respect to t, where t is a parameter that parametrizes the curve.
First, let's find the derivative of y with respect to x:
dy/dx = d(3x^2 - 2x + 5)/dx = 6x - 2
Next, we'll square this derivative:
(dy/dx)^2 = (6x - 2)^2 = 36x^2 - 24x + 4
The length of the curve can now be calculated using the definite integral:
L = ∫[0, 2] sqrt(1 + (dy/dx)^2) dx
L = ∫[0, 2] sqrt(1 + (36x^2 - 24x + 4)) dx
L = ∫[0, 2] sqrt(36x^2 - 24x + 5) dx
To solve this integral, we can use a change of variables. Let u = 6x - 2, then du = 6 dx:
L = (1/6) ∫[0, 2] sqrt(u^2 + 5) du
Next, we can use a trigonometric substitution, let u = √5 tanθ, then du = √5 sec^2θ dθ:
L = (1/6) ∫[0, θ] √(5tan^2θ + 5) √5 sec^2θ dθ
L = (1/6) ∫[0, θ] √(5sec^2θ) √5 sec^2θ dθ
L = (1/6) ∫[0, θ] 5 sec^3θ dθ
Since u = √5 tanθ, we can substitute back:
L = (1/6) ∫[0, θ] 5(√(5/5) sec^3θ) dθ
L = (1/6) ∫[0, θ] 5(√(5/5) secθ tanθ sec^2θ) dθ
L = (5√5/6) ∫[0, θ] secθ tanθ sec^2θ dθ
Using the property that (d/dθ) secθ = secθ tanθ, we can simplify further:
L = (5√5/6) ∫[0, θ] secθ (d/dθ)(secθ) dθ
L = (5√5/6) ∫[0, θ] d(sec^2θ)
L = (5√5/6) [sec^2θ] [0, θ]
L = (5√5/6) (sec^2θ - sec^2(0))
Since sec^2(0) = 1, we have:
L = (5√5/6) (sec^2θ - 1)
By evaluating the integral at the upper limit of integration, which is θ, we can find the final length of the curve.
b) To determine the length of the curve over the interval [0, 2] using the distance formula, we can consider the curve as a function y = f(x). First, we need to find the derivative of y with respect to x:
dy/dx = d(3x^2 - 2x + 5)/dx = 6x - 2
The length of the curve can be calculated using the formula:
L = ∫[a, b] sqrt(1 + (dy/dx)^2) dx
L = ∫[0, 2] sqrt(1 + (6x - 2)^2) dx
To solve this integral, we can use the substitution method. Let u = 6x - 2, then du = 6 dx:
L = (1/6) ∫[0, 2] sqrt(1 + u^2) du
This integral can be evaluated using trigonometric substitution, but it can get quite complicated. Therefore, an alternative approach is to use numerical methods or calculators to approximate the integral value. For example, using numerical integration techniques like Riemann sums or Simpson's rule, we can estimate the length of the curve.