You are given the four points in the plane A = (5,-6), B = (9,1), C = (14,-6), and D = (19,2). The graph of the

function f(x) consists of the three line segments AB, BC and CD. Find the integral f(x)dx from 5 to 19 by interpreting the integral in terms of sums and/or differences of areas of elementary figures.

To find the integral of f(x)dx from 5 to 19, we can interpret it as the sum (or difference) of areas of elementary figures. In this case, we will break down the area under the curve into three regions corresponding to the three line segments AB, BC, and CD.

The first step is to find the equations of the three line segments: AB, BC, and CD.

Using the point-slope form of a line (y - y1) = m(x - x1), we can find the equations of each line segment:

1. AB:
- Point A: (5, -6)
- Point B: (9, 1)

The slope m_AB = (y2 - y1) / (x2 - x1) = (1 - (-6)) / (9 - 5) = 7 / 4

Using the point-slope form with point A:
y - (-6) = (7/4)(x - 5)
y + 6 = (7/4)x - 35/4
y = (7/4)x - 59/4

2. BC:
- Point B: (9, 1)
- Point C: (14, -6)

The slope m_BC = (y2 - y1) / (x2 - x1) = (-6 - 1) / (14 - 9) = -7 / 5

Using the point-slope form with point B:
y - 1 = (-7/5)(x - 9)
y = (-7/5)x + 16/5

3. CD:
- Point C: (14, -6)
- Point D: (19, 2)

The slope m_CD = (y2 - y1) / (x2 - x1) = (2 - (-6)) / (19 - 14) = 8 / 5

Using the point-slope form with point C:
y - (-6) = (8/5)(x - 14)
y + 6 = (8/5)x - 112/5
y = (8/5)x - 142/5

Now that we have the equations of the line segments, we can find the areas of the elementary figures and sum them up to find the integral.

1. Area under AB:
The area under AB can be represented by a trapezoid. The base of the trapezoid is 9 - 5 = 4 units, and the height is the difference in y-coordinates at the endpoints, which is 1 - (-6) = 7 units.

The area of the trapezoid is given by the formula: Area = (base1 + base2) * height / 2

Plugging in the values for the base and height:
Area_AB = (4 + 4) * 7 / 2 = 28 / 2 = 14 square units

2. Area under BC:
The area under BC can be represented by a rectangle. The base of the rectangle is 14 - 9 = 5 units, and the height is the difference in y-coordinates at the endpoints, which is (-6) - 1 = -7 units (we take the absolute value, so it becomes 7 units).

The area of the rectangle is given by the formula: Area = base * height

Plugging in the values for the base and height:
Area_BC = 5 * 7 = 35 square units

3. Area under CD:
The area under CD can be represented by a trapezoid. The base of the trapezoid is 19 - 14 = 5 units, and the height is the difference in y-coordinates at the endpoints, which is 2 - (-6) = 8 units.

The area of the trapezoid is given by the formula: Area = (base1 + base2) * height / 2

Plugging in the values for the base and height:
Area_CD = (5 + 5) * 8 / 2 = 50 / 2 = 25 square units

Now, we sum up the areas of the elementary figures:
Integral of f(x)dx from 5 to 19 = Area_AB + Area_BC + Area_CD = 14 + 35 + 25 = 74 square units