Problem 17 - Essential Calculus

Question

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ y=x, \quad y=2-x, \quad y=0 $$

Step-by-Step Solution

Answer

The area of the region bounded by the given functions is 1 square unit.

Step 1: Sketching the Graphs

Begin by sketching the graphs of each of the three functions. The graph \(y=x\) is a straight line passing through the origin with slope 1. The graph \(y=2-x\) is a straight line with y-intercept 2 and slope -1. Lastly, \(y=0\) represents the x-axis. Observe that these three lines form a triangle on the Cartesian plane.

Step 2: Finding the Intersection Points

The intersection points of the graphs represent the vertices of the triangle. Find these by setting the equations equal to each other. Setting \(y=x\) and \(y=2-x\) equal to each other gives \(x=2-x\), from which \(x=1\) is derived. Therefore, the lines intersect at the point (1,1). The line \(y=x\) intersects the x-axis at the origin (0,0), and the line \(y=2-x\) intersects the x-axis at the point (2,0).

Step 3: Calculating the Area

The area of the region bounded by the graphs can be found using the formula for the area of a triangle, which is \(\frac{1}{2} \times \text{base} \times \text{height}\), or using integral calculus. In this case, the base of the triangle is the line segment from (0,0) to (2,0), so the base is 2. The height of the triangle is the line segment from (1,1) to (1,0), so the height is 1. Thus, the area of the triangle is \(\frac{1}{2} \times 2 \times 1 = 1\).