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Chapter 5: Problem 17

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

Short Answer

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

Step by step solution

01

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.
02

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).
03

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 \(\frac{1}{2} \times 2 \times 1 = 1\).

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Most popular questions from this chapter

Find the accumulation function \(F\). Then evaluate \(F\) at each value of the independent variable and graphically show the area given by each value of \(F\). $$ F(y)=\int_{-1}^{y} 4 e^{x / 2} d x \quad \text { (a) } F(-1) \quad \text { (b) } F(0) \quad \text { (c) } F(4) $$

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=x^{4}-2 x^{2}, \quad y=2 x^{2} $$

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(x)=x^{2}-4 x, \quad g(x)=0 $$

The management at a certain factory has found that a worker can produce at most 30 units in a day. The learning curve for the number of units \(N\) produced per day after a new employee has worked \(t\) days is \(N=30\left(1-e^{k t}\right)\). After 20 days on the job, a particular worker produces 19 units. (a) Find the learning curve for this worker. (b) How many days should pass before this worker is producing 25 units per day?

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=x^{4}-4 x^{2}, g(x)=x^{3}-4 x $$
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