Math Multitool

Don’t you dare laugh at me when in my last post I showed an illustration of an integral to calculate the area of a square. Because that’s math: a toolbox with multitools. To make it even worse, here’s an example of integration to calculate the area of a trapezoid.

This is a piece of cake. Height is e, base is d. The trapezoid consists of two equal triangles on both sides and a rectangle in the middle.
The area follows from this:

    \[2\cdot\tfrac{1}{2}be+(c-b)e=be+ce-be=ce\]

Now we’re gonna check this by integration. The left side can be defined as the function f(x)=\tfrac{e}{b}x and the top side as f(x)=e as it is constant. The right side is the mirror image of the left triangle, so we simply take the first integral (of the left triangle) twice.

    \[2\int_0^b \tfrac{e}{b}x\mathrm{d}x+\int_b^c e\mathrm{d}x\]

We get

    \[2\left[\tfrac{e}{2b}x^2\right]_0^b + \left[ex\right]_b^c\]

or

    \[\left[\tfrac{e}{b}x^2\right]_0^b + \left[ex\right]_b^c\]

which results in

    \[\tfrac{eb^2}{b}-0 + ec-eb\]

and

    \[eb+ec-eb=ec\]

again.

Math = Multitool. Remember.