Final answer:
The integral to set up for the arc length of the curve f(x) = e⁴¹ from x = 1 to x = 10 is ∫₁¹√(1 + (4x³e⁴¹)²) dx.
Step-by-step explanation:
The student is asking to set up, but not evaluate, the integral for the arc length of the curve represented by the function f(x) = e⁴¹ from x = 1 to x = 10.
To find the arc length, we use the formula for the arc length s of a function y = f(x), which is given by the integral from a to b of √(1 + (dy/dx)²) dx. For this function, first, we need to calculate the derivative of f'(x), which is f'(x) = 4x³e⁴¹.
Then, we square this derivative. Adding 1 to the square of the derivative and taking its square root gives us √(1 + (4x³e⁴¹)²). Therefore, the integral to set up for the arc length from x = 1 to x = 10 is ∫₁¹√(1 + (4x³e⁴¹)²) dx.
The complete question is:Consider the curve given by f(x)=eˣ⁴⁺¹SET UP (DO NOT EVALUATE) the integral for the are-length of the curve for 1≤x≤10. is: