To find the exact length of the curve given by the equation x = (1/3)y^3 + 16y + 25, we will use the formula for the length of a curve defined by a function:
L = ∫ab √(1 + (dy/dx)²) dy
First, we need to express everything in terms of y rather than x. Let’s differentiate x with respect to y to find dx/dy:
dx/dy = y² + 16
Next, we compute ((dx/dy)²):
((dx/dy)²) = (y² + 16)²
Now we can use the formula for the length of the curve:
First we need to calculate 1 + (dy/dx)². To do this, we need to find dy/dx, which is the reciprocal of dx/dy:
dy/dx = 1 / (y² + 16)
Thus,
((dy/dx)²) = 1 / (y² + 16)²
Now plug this into our curve length integral:
L = ∫ab √(1 + 1 / (y² + 16)²) dy
Next, determine the limits of integration (a, b). Depending on the problem, these may correspond to specific points on the y-axis through which the curve passes.
Once we have our limits and set up the integral, we can compute the integral to find the exact length of the curve.
Note: Evaluating the resulting integral may require numerical integration techniques if it does not simplify well.