To approximate the tangent to a curve at a point (the derivative), find the slope of the line that connects closely-spaced points on either side (the symmetric difference quotient):

$m=(y_1-y_{–1})/(x_1-x_{–1})$

The tangent line:

$y=m(x-x_0)+y_0$

To approximate the area under a curve (the integral), partition it into narrow trapezoids and find the sum of the areas (the method of exhaustion).

The area of each trapezoid is the area of the rectangle plus the area of the triangle on top.
It is equivalent to the width × the average of the heights:

$A\approx (x_1-x_0)((y_1+y_0)/2)$

The arc length is approximated by the diagonal:

$s\approx \sqrt{(x_1-x_0{)}^2+(y_1-y_0{)}^2}$

The narrower the trapezoids, the closer the approximation.

To approximate area using polar coordinates, partition the figure into narrow triangles.

If the incremental angle ${\theta }_1-{\theta }_0=\alpha$ is very small, the triangles may be treated as if they were right triangles with both hypotenuse and long side of length $r$, and short side of length $r\text{sin}\alpha$.

The area of each right triangle would be approximately $(r\cdot r\text{sin}\alpha )/2$.

The area of the segment of the figure may be more closely approximated by averaging the areas of the $r_0$ and $r_1$ right triangles, as with the trapezoids above; for very small α,

$A\approx (r_0{r}_1\text{sin}\alpha )/2$

The arc length is approximated by the $r_0–r_1$ diagonal:

$s\approx \sqrt{(r_1-r_0{)}^2+r_0{r}_1(\text{sin}\alpha {)}^2}$

or, in Cartesian form:

$s\approx \sqrt{(r_1\text{cos}{\theta }_1-r_0\text{cos}{\theta }_0{)}^2+(r_1\text{sin}{\theta }_1-r_0\text{sin}{\theta }_0{)}^2}$

The narrower the angle, the closer the approximation.

Begin with a parabola, $y=x^2$.

Let $x_0=$ 0.4.  The slope of the tangent at $x_0$ is the derivative

${\displaystyle \frac{d}{dx}{x}^2=2x_0=}$ 0.8

Using an x increment of 0.000001, the slope of the line connecting points on either side is 0.800000000003.

Let $x_1=$ 1.1.  The area under the curve from 0 to $x_1$ is the integral

${\displaystyle {\int }_0^{x_1}{x}^{2}dx=\frac{{x_1}^3}{3}=}$ 0.443667

The calculated area by trapezoids is 0.443668.

Now for an ellipse, where $a=16,b=5,c=\sqrt{a^2-b^2},$ and ${\displaystyle e=\frac{c}{a}.}$
When the origin is at one focus, ${\displaystyle r=\frac{a(1-e^2)}{1-e\text{cos}\theta }}$

Let θ₀ = 5°. The slope of the tangent at θ₀ is

${\displaystyle \frac{–b^2(x-c)}{a^{2}y}=}$ -0.5309683222

Using an angle increment of 0.000001 radians, the slope of the line connecting points on either side is -0.5309683217.

The area of an ellipse is $\pi ab=$ 251.327412
The calculated area by triangles is 251.327413.

The circumference of an ellipse has no elementary solution, but may be calculated to any degree of precision by the following formula:

${\displaystyle C=\pi (a+b)[1+\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{(2n-1)!!}{2^{n}n!}\right)}^2\frac{h^n}{(2n-1{)}^2}],}$ where ${\displaystyle h=(a-b{)}^2/(a+b{)}^2}$

Taken to greater than 15 significant figures, the circumference of the ellipse is 70.5824241.
The calculated circumference by $r_0–r_1$ segments is 70.5824236.

Note that infinitesimal methods may be used to find the tangent to any ellipse, such as one rotated at an arbitrary angle; and they may be used to find any segment of area or circumference.

Now for something a lot more practical. What is the area of the shaded slice of pizza?

The pizza radius is 1 and the slice dimensions are 1⁄3 × 1⁄3.

The equation for the top half of the pizza is $y=\sqrt{1-x^2}.$

Using an x increment of 0.000001, the calculated area by trapezoids is 0.06404, or 58 % of a full slice.

The length of the crust by $r_0–r_1$ segments is 0.38989.

Starting the year in which the owner turns 72, retirement accounts must make an annual required minimum distribution based on life expectancy tables published by the Internal Revenue Service. The graph at right shows what happens to a $100,000 account with the specified growth rate as RMD’s are withdrawn in monthly installments.

growth rate:  4.0 %

age:  84 |  month:  February

The rate of change in February the year the owner turns 84 is $ -1,852 per year.
The area under the curve (total value × time) is $ 2,489,786.