To find the arc length function for the curve given by y = 2x^{3/2} between the points P(0, 36) and (4, 32), we begin with the formula for arc length:
L = ∫_a^b ext{√(1 + (dy/dx)²) dx}
where dy/dx is the derivative of the function with respect to x, and [a, b] are the limits of integration corresponding to the x values of the endpoints.
Step 1: Find the derivative
The function can be differentiated to find dy/dx:
y = 2x^{3/2}
Using the power rule of differentiation:
dy/dx = 2 * (3/2)x^{1/2} = 3x^{1/2}
Step 2: Calculate (dy/dx)²
(dy/dx)² = (3x^{1/2})² = 9x
Step 3: Substitute into the arc length formula
Now substitute into the arc length integral:
L = ∫_0^4 √(1 + 9x) dx
Step 4: Simplify the integral
The square root in the integral can be simplified:
∫_0^4 √(1 + 9x) dx
This is a standard integral and typically requires substitution. Let:
u = 1 + 9x ⇒ du = 9 dx or dx = du/9
When x = 0, u = 1, and when x = 4, u = 37.
Now substitute:
L = (1/9) ∫_1^37 √u du
Step 5: Solve the integral
The integral of √u is:
(2/3)u^{3/2}.
Thus:
L = (1/9) * (2/3) [u^{3/2}]_1^37
Plugging in the limits:
L = (2/27) (37^{3/2} - 1^{3/2})
Final Arc Length Function
After calculating the above expression, we arrive at:
L = (2/27) (37√37 - 1)
This represents the total arc length of the curve from the starting point P(0, 36) to the ending point (4, 32). It’s important to note that this method provides the precise length of the curve segment between the specified points.