Archimedean Triangles and Quadrature of a Parabola

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We take the points at x locations x_0, x_0+h/2, and x_0+h on the curve Y=X^2, and we create triangles by connecting the points and their tangents. Note that each triangle has an area independent of x_0.

If we wish to find the area between chord CD and the parabola, we can do this by repeating the process with the smaller triangles that straddle the curve. If we treat those triangles as CDH, we get a geometric sequence, and we get A=h^3/6.

Expressions

Name Input
Bottom Area Derive Input Maple Input MathML Input Mathematica Input Maxima Input Mupad Input TI-Nspire Input text Input Image
Top Area Derive Input Maple Input MathML Input Mathematica Input Maxima Input Mupad Input TI-Nspire Input text Input Image
Right Area Derive Input Maple Input MathML Input Mathematica Input Maxima Input Mupad Input TI-Nspire Input text Input Image
Left Area Derive Input Maple Input MathML Input Mathematica Input Maxima Input Mupad Input TI-Nspire Input text Input Image