Originally Posted by
eumyang First, find the equations of the three lines that go through these points.
L1 -> the line going through (0, 0) and (-1, 2)
L2 -> the line going through (0, 0) and (10, 1)
L3 -> the line going through (-1, 2) and (10, 1)
Put all equations in slope-intercept form (y = mx + b). I'm going to call these three functions $\displaystyle L_1(x), L_2(x), L_3(x)$ respectively.
Draw the triangle. From x = -1 to x = 0, you can find the area of the part of the triangle to the left of the y-axis by finding the integral of the difference of the two functions L3(x) - L1(x):
$\displaystyle A = \displaystyle \int_{-1}^0 L_3(x) - L_1(x) \,dx \ldots$
Then, from x = 0 to x = 10, you find the area of the part of the triangle to the right of the y-axis by finding the integral of the difference of the two functions L3(x) - L2(x):
$\displaystyle \ldots + \displaystyle \int_{0}^{10} L_3(x) - L_2(x) \,dx $
So the complete integral will be
$\displaystyle A = \displaystyle \int_{-1}^0 L_3(x) - L_1(x) \,dx + \displaystyle \int_{0}^{10} L_3(x) - L_2(x) \,dx$