TITLE: Finding the area of triangle given the vertices with integration
QUESTION [1 upvotes]: I'm trying to find the area of the triangle given the vertices $(0,1)$, $(2,6)$, $(7,10)$.
I have to use integration, given these formulas to use it
$$m = \frac{y_2-y_1}{x_2-x_1}$$
$$y-y_1=m(x-x_1)$$
Using what I know Ive gotten these integrals
$$\int_0^2 \frac52x - \frac97x \,dx + \int_7^2 \frac45x - \frac97x \,dx $$
The right answer is $\frac{17}2$ but this does not give me it.
I am unsure where to use the second formula.

REPLY [1 votes]: There is an error in the integrand of the second integral. The correct expression is
$$I=\int_0^2 (y_1-y_3)dx + \int_2^7 (y_2-y_3)dx$$
where the three lines are
$$y_1=\frac 52 x+1$$
$$y_2=\frac 45 x+\frac{22}{5}$$
$$y_3=\frac 97 x+1$$
So, the integral should be
$$I=\int_0^2 \left(\frac52x - \frac97x\right) \,dx + \int_2^7 \left(\frac45x - \frac97 x +\frac{17}{5}\right)\,dx =\frac{17}{7}+\frac{85}{14}=\frac{17}{2}$$
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