# Finding the area of triangle given the vertices with integration

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.

## 1 Answer

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 $$I=\int_0^2 \left(\frac52x - \frac97x\right) \,dx + \int_2^7 \left(\frac45x - \frac97 x +\frac{17}{5}\right)\,dx =\frac{17}{7}+\frac{85}{14}=\frac{17}{2}$$

• Thanks I had trouble finishing this one and this showed me where I was wrong with it, pretty close thise – user708741 Sep 26 '19 at 13:54