Final answer:
To calculate the arc length of y = 3x + 1 over the interval [0,10], we can use the formula Arc Length = ∫ √(1 + (dy/dx)²) dx. By finding the derivative dy/dx and substituting it into the formula, we can evaluate the integral and calculate the arc length.
Step-by-step explanation:
To calculate the arc length of y = 3x + 1 over the interval [0,10], we can use the formula:
Arc Length = ∫ √(1 + (dy/dx)²) dx
First, we need to find dy/dx (the derivative of y with respect to x):
dy/dx = d(3x + 1)/dx = 3
Next, we substitute the derivative into the formula:
Arc Length = ∫ √(1 + 3²) dx = ∫ √(1 + 9) dx = ∫ √10 dx
To integrate this, we can set up a trigonometric substitution:
Let u = √10x, then du = √10 dx
Substituting back into the integral:
Arc Length = (1/√10) ∫ √u du
Integrating, we get:
Arc Length = (1/√10) (2/3)u^(3/2) + C
Finally, evaluate the arc length from 0 to 10:
Arc Length = (1/√10) (2/3)(√10 × 10)^(3/2) - (1/√10) (2/3)(√10 × 0)^(3/2) = (2/3)(√10 × 10)^(3/2)