Question

In Exercises 1 and 2 , find the distance between the points using (a) the Distance Formula and (b) integration. $$ (1,2), \quad(7,10) $$

Short answer

The distance between the points (1,2) and (7,10) is 10 units computed by both the Distance formula and integration.

Step-by-step solution

Distance Formula

First calculate the distance using the Distance Formula. The Distance Formula is given by \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). Substitute \(x_1 = 1, y_1 = 2, x_2 = 7, y_2 = 10\). Thus \(d = \sqrt{(7-1)^2 + (10-2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\).

Integration

Now calculate the distance between the points using the method of integration. We obtain the equation of the line passing through these two points and then integrate the differential arc length. The equation of the line that passes through the points \((1,2)\) and \((7,10)\) is \(y = x + 1\). The differential arc length is \(ds = \sqrt{1 + (dy/dx)^2} dx\). Substitute \(dy/dx = 1\) because the derivative of \(y = x + 1\) is 1. So \(ds = \sqrt{1+1} dx = \sqrt{2} dx\). Now integrate over the interval from \(x = 1\) to \(x = 7\) to find \(d = \int_{1}^{7} {\sqrt{2} dx} = \sqrt{2}\times(7 - 1) = \sqrt{2} \times 6 = \sqrt{24} \approx 10\). Note that the result is not exactly 10 due to the approximation of the square root of 24.