Problem 32 - Calculus Early Transcendentals

Question

Consider the segment of the line \(y=m x+c\) on the interval \([a, b] .\) Use the arc length formula to show that the length of the line segment is \((b-a) \sqrt{1+m^{2}}\) Verify this result by computing the length of the line segment using the distance formula.

Step-by-Step Solution

Answer

Answer: The length of the line segment is \((b-a) \sqrt{1+m^2}\).

Step 1: Compute the derivative of the function

The function is given as \(f(x) = y = mx + c\). To find its derivative, we simply take the derivative of the function with respect to x: \(\displaystyle \frac{dy}{dx} = \frac{d(mx + c)}{dx} = m\)

Step 2: Plug the derivative into the arc length formula

Now that we have the derivative, plug it into the arc length formula: \(\displaystyle L = \int_{a}^{b} \sqrt{1+(\frac{dy}{dx})^2} dx = \int_{a}^{b} \sqrt{1+m^2} dx\)

Step 3: Integrate the expression

Since \(\sqrt{1+m^2}\) is a constant, we can take it out of the integral: \(\displaystyle L = \sqrt{1+m^2} \int_{a}^{b} dx\) Now, integrate the expression: \(\displaystyle L = \sqrt{1+m^2} (x \bigg|_{a}^{b})\) \(\displaystyle L = \sqrt{1+m^2} (b-a)\)

Step 4: Compute the length using the distance formula

To verify the result, we will compute the length of the line segment using the distance formula. First, find the coordinates of the endpoints on the interval \([a, b]\): \((a, f(a)) = (a, ma + c)\) \((b, f(b)) = (b, mb + c)\) Now, apply the distance formula: \(\displaystyle D = \sqrt{(b-a)^2 + ((mb+c) - (ma+c))^2}\) \(\displaystyle D = \sqrt{(b-a)^2 + (mb-ma)^2}\) \(\displaystyle D = \sqrt{(b-a)^2(1+m^2)}\) \(\displaystyle D = (b-a) \sqrt{1+m^2}\) This confirms that the length of the line segment computed using both methods is indeed \((b-a) \sqrt{1+m^2}\).

Key Concepts

Line Segment Length

When we talk about the length of a line segment, we are referring to the distance between two points on a coordinate plane. These two points are the endpoints of the segment. Understanding the length of a line segment is fundamental to geometry, and it serves as a building block for more complex calculations in many fields, including architecture, engineering, and computer graphics.

For a straight line segment, finding this length is relatively straightforward. If we have coordinates for both endpoints, we can apply the distance formula to determine the length. This becomes especially handy when dealing with line segments on a graph.

Integral Calculus

Integral calculus is a branch of mathematics that deals with the accumulation of quantities and the area under a curve. When it comes to finding the arc length of a function between two points, integral calculus is the go-to tool. What we essentially do is sum an infinite number of infinitesimally small line segments along the curve to find the total length.

By integrating a certain function derived from the original function we are interested in, we can find areas, volumes, and, relevant to our discussion, lengths of curves. This aspect of calculus is vital for many scientific and engineering disciplines where such quantities need to be precisely determined.

Derivative of a Function

The derivative of a function is a measure of how a function's output value changes as its input value changes. In simpler terms, it tells us the rate at which one quantity is changing with respect to another. In the context of the arc length formula, the derivative provides us with an essential piece of information—the slope of the tangent to the curve at any given point.

When we calculate the arc length over an interval, the derivative's square is used within the arc length formula to account for both vertical and horizontal changes as we move along the curve. In the case of a line, the derivative is constant, making the integration process a bit easier, as seen in our exercise solution.

Distance Formula

The distance formula is derived from the Pythagorean theorem and allows us to calculate the distance between two points in a coordinate system. This formula is crucial when it comes to finding the length of a line segment, especially when the segment is a straight line. By knowing the coordinates of the endpoints, we can plug them into the distance formula to get the exact length of the segment in question.

It's interesting to note that the distance formula can also be derived from the arc length formula by considering a linear function. This relationship shows the conceptual link between these fundamental tools in mathematics, bridging the gap between geometry and calculus.