help with integrals?

Question

You are given the four points in the plane A=(-6,-8) B= (-2,1) C=(3,-7) and D=(8,8)
The graph of the function f(x) consists of the three line segments AB, BC and CD. Find the integral f(x) dx by interpreting the integral in terms of sums and/or differences of areas of elementary figures.

integral 8(top) -6 (bottom) f(x) dx=??

I found the x intercepts to be -11/8 and 16/3
I can't seem to get the right answer though.

Answer 1

The area is the sum of 4 triangular areas.

There are 3 intercepts:
A-B: y = (9/4)x + 11/2 so x = -22/9 at y = 0
B-C: y = (-8/5)x - 11/5 so x = -11/8 at y = 0
C-D: y = 3x - 16 so x = 16/3 at y = 0

First area is formed from x axis down to the line A-B.
A1 = (1/2)(-22/9 + 6)(8) = 4(32/9) = 128/9

Second area is formed at B down to the x axis.
A2 = (1/2)(-11/8 + 22/9)(1) = 77/144

Third area is from x axis down to BC from y=0 to C and from the x axis down to CD from C.
A3 = (1/2)(16/3 + 11/8)(7) = 1127/48

Last area is from the x axis up to CD.
A4 = (1/2)(8 - 16/3)(8) = 32/3

Area = A1 + A2 + A3 + A4
Area = 128/9 + 77/144 + 1127/48 + 32/3
Area = (128/3 + 77/48 + 1127/16 + 32)/3
Area = 7042/144 = 48.9

If I did all the math right, that's the area.

Answer 2

There is a third x-intercept at -22/9 where the segment AB hits the axis. Let Int[a,b]f(x)dx denote the integral of f(x) from a (lower limit) to b. The work looks like this:

Int[-6.-2]((9/4)*x + 11/2)dx + Int[-2,3]((-8/5)*x - 11/5)dx +
Int[3,8](3x - 16)dx = ((9/4)*(x^2)/2 + (11/2)*x)[-6,-2] +
((-8/5)*(x^2)/2 - (11/5)*x)[-2.3] + ((3/2)*x^2 - 16x)[3,8] =
(9/8)*(-4) + (11/2)*4 - (4/5)*5 - (11/5)*5 +(3/2)*55 - 16*5 =
- 53/2 .

For the geometric check, let E be the x-intercept at -22/9, F be -11/8, and G be 16/3. Also, "drop" a perpendicular from A to the x-axis and let H be the "foot" from A, and let J be the foot from D. Now the triangle AHE has base 32/9 and height 8, for area 128/9 (and this is negative since the region is below the x-axis). Furthermore, EBF has base 4/9 + 5/8 and height 1, for area 77/144; ECG has base 161/24 and height 7, so area - 1127/48; and GDJ has base 8/3 and height 8 for area 32/3. Finally, -128/9 + 77/144 - 1127/48 + 32/3 = 53/2.