What you are looking at
Every strip is a thin rectangle (or a trapezoid). Its area is width × height – the width is the small step along the bottom, the height is the value of the curve there. Add all the strips together and you have the area under the curve. That is the whole idea of an integral. The symbol
b ∫ f(x) dx a
says exactly this: ∫ is a long S for sum, f(x) is the height of a strip, dx is its (very small) width, and a and b are where you start and stop adding.
With a few strips the answer is rough. With more strips it gets better, and the "exact" integral is what you would get with infinitely many infinitely thin strips. Try the four ways of choosing the height: the middle and the trapezoid are much better than the edges, which is why computers use them.
The trick with units
The area always has the unit of the height times the unit of the width. Speed [m/s] × time [s] = distance [m]. Load [kN/m] × length [m] = force [kN]. Power [kW] × time [h] = energy [kWh]. If you know what the axes mean, you know what the integral means.
The whole story, with a bit of history and examples from engineering, is in my note Integrals Without Fear. The Beam Calculator uses exactly these trapezoids to get the deflection of a beam.