By Richard. Abbatt
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Additional info for A treatise on the calculus of variations
Remember all the sums involved: (v(1) × ¨t) + (v(1+¨t) × ¨t) + (v(1+2¨t) × ¨t) + … + (v(2 – ¨t) × ¨t, as ¨t becomes increasingly smaller. Then we can ¿nd the distance traveled knowing that p (t ) t 2 by subtraction. Because p (2) 4 and p (1) 1 , the car traveled 3 miles. We can consider other pairs of time values. The summing process of the integral will always yield the same result as just subtracting the positions, because all these processes are referring to the same scenario of a moving car.
If the function is below the axis, then the integral is negative. If the function is part above the axis and part below, the integral combines the two. It’s easy. When the graph goes below the axis, the integral is negative; when above, positive. The summation fact, a to b plus b to c equals The Fundamental a to c, works regardless of whether the graph Theorem of Calculus goes above or below the axis. ³ b F c( x)dx F (b) F (a) Integrals behave “opposite” of derivatives graphically. For a function f (x), we can de¿ne a function F (x) as the integral of f from starting time a to ending time x.
There is a whole different world of ideas associated with non-Euclidean geometries in which such formulas are not as simple. 47 Archimedes and the Tractrix Lecture 9 Today we’re going to look at a couple of examples of this strategy that preceded the de¿nition of the integral, and then some examples that actually are very modern. I Lecture 9: Archimedes and the Tractrix n the 17th century, Bonaventura Cavalieri analyzed shapes using his method of indivisibles. If one thinks of the surface of a sphere as comprised of many tiny triangles, then the volume of the sphere can be viewed as made up of many tiny tetrahedra with those triangles as bases and the center of the sphere as the top of each tetrahedron.
A treatise on the calculus of variations by Richard. Abbatt