Answer
Don't panic at the instructions "derive the area formula." All this means is we need to use integration to find the area of the triangle, and we should get
![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_166.png)
when we evaluate the integral at the end.
This problem is the same as problem 2 above except that we have letters instead of numbers for the side lengths.
Let y be the height of the slice at position x:
![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_graphik_46.png)
We have similar triangles again. The big triangle has height h and base b; the smaller triangle has height y and base x:
By similarity, we know that
![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_167.png)
Since x is the variable of integration, we need to get the height (y) of the slice in terms of x:
![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_168.png)
The area of the slice at position x is
![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_169.png)
The variable x goes from 0 at the right tip of the triangle to b at the left edge of the triangle.
Taking the integral, we find that the area of the triangle is
![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_170.png)
Let's evaluate this integral and see what we get:
![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_171.png)
We found the area of the triangle using an integral, evaluated the integral, and got the formula for the area of a triangle. Finally, the long-awaited derivation of for the area of a triangle is ours. We have the power, so we might as well use it.