Question

27 - 30. area versus net area graph the following functions. then use geometry (not riemann sums) to find the area and the net area of the region described.

1. the region between the graph of ( y = - 3x ) and the ( x ) - axis, for ( - 2 leq x leq 2 )

Explanation:

Step1: Find intersection points

When \(y = 0\), \(x = 0\).

Step2: Calculate area of two triangles

For \(x\in[-2,0]\), \(y=-3x\geq0\). The base of the triangle is \(b_1=\vert - 2\vert=2\), the height is \(h_1=\vert y(-2)\vert=\vert-3\times(-2)\vert = 6\). Area of this triangle \(A_1=\frac{1}{2}\times2\times6 = 6\).
For \(x\in[0,2]\), \(y=-3x\leq0\). The base of the triangle is \(b_2 = 2\), the height is \(h_2=\vert y(2)\vert=\vert-3\times2\vert=6\). Area of this triangle \(A_2=\frac{1}{2}\times2\times6 = 6\).

Step3: Calculate total area

Total area \(A=A_1 + A_2=6 + 6=12\).

Step4: Calculate net area

Net area \(A_{net}=A_1 - A_2=6-6 = 0\).

Answer:

The area is \(12\) and the net area is \(0\).