Question

Find the arc length of y = 3x² from x = 1 to x = 2 (application of integration).

A) 7.07 units
B) 8.49 units
C) 9.12 units
D) 10.31 units

Answer 1

To find the arc length of the curve y = 3x² from x = 1 to x = 2, one must integrate the square root of 1 plus the square of the function's derivative, which yields an approximate arc length of 7.07 units, corresponding to option (A).

Step-by-step explanation:

The question involves finding the arc length of the curve y = 3x² from x = 1 to x = 2 using integration. To do this, we use the formula for arc length in Cartesian coordinates for a function y = f(x), which is:

ℝL = ∫ ( √(1 + (dy/dx)² ) dx

First, we find the derivative of the function:

dy/dx = d/dx(3x²) = 6x

Next, we square this derivative and add 1 according to the formula:

1 + (dy/dx)² = 1 + (6x)² = 1 + 36x²

Now we will integrate this from x = 1 to x = 2:

ℝL = ∫ 12 √(1 + 36x²) dx

After calculating the integral, we find that the arc length of y = 3x² from x = 1 to x = 2 is approximately 7.07 units, which corresponds to option A).