Final answer:
To calculate the length of the linear curve y = 3x - 2 from x = -3 to x = 1, we use the arc length formula for a function, solve its derivative, which is constant, and integrate over the given interval, resulting in the arc length being 4√10 units.
Step-by-step explanation:
To find the length of the curve y = 3x − 2, for x in the interval −3 ≤ x ≤ 1, we will need to use calculus, specifically the arc length formula for a function. Contrary to the information provided about circular arcs, this is a linear function, not a segment of a circle, so the radius of curvature is not applicable here. Instead, the formula for the arc length S of a function y = f(x) from a to b is given by:
S = ∫ab √(1 + (dy/dx)2)dx
First, we differentiate the function with respect to x, which gives us dy/dx = 3 since the function is linear. The derivative is constant, so the integrand simplifies to √(1 + 32) = √(1 + 9) = √10. As dy/dx is constant, the integration from x = -3 to x = 1 is straightforward:
S = ∫−31 √10 dx
This integral represents the multiplication of √10 by the length of the interval on the x-axis, which is 4 units (from −3 to 1). Hence, the arc length is:
S = 4 √10
Thus, the length of the curve from x = −3 to x = 1 is 4 √10 units. Remember that the arc length for this linear function does not rely on the radius of curvature, unlike a circular arc.