Area and right-endpoint height

Parametric equations

The right endpoint moves along this curve:

The area from $0$ to $x$ has a closed form:

So $A^{\prime }(x)=f(x)$. Past the right endpoint, take one more piece of width $h$, and write $\Delta A=A(x+h)-A(x)$. As $h$ approaches $0$, $\Delta A/h$ approaches $f(x)$. The top of the strip is horizontal, and its height is $f(x)$ at the start of that piece.

Adding a constant to an antiderivative does not change the difference of the two ends. If the lower limit were the one that moved, the height at that end would take a minus sign. That is a consequence of the same formula. The figure moves only the upper limit.

Interaction

• Right endpoint x: move the upper limit; the gold region extends to the new right end
• Strip width h: change the width of the rectangle at the right end; $\Delta A/h$ moves toward or away from the height of the curve

What to notice

• On this curve, the rate of change of the area is the height at the right endpoint.
• While the strip still has width, $\Delta A/h$ and $f(x)$ can stand apart. They meet only as the width approaches $0$.
• The curve leaves the top-left corner of the strip and runs to the right. Where this piece rises, it shows above the strip, and where it falls, it drops into the strip.

Related

• Riemann sum
• Tangent approximation
• Geometric definition of the natural logarithm
• Limits and Riemann sums

Further reading

• Fundamental theorem of calculus (Wikipedia)
• Derivative (Wikipedia)