Que es una notacion de sumatoria en calculo integral?

Primero pongamos la definición

<img src="https://tex.z-dn.net/?f=%5Ctextbf%7BD%7D%5Ctextbf%7Befinici%5C%27on.%20%7D%5Ctext%7BLa%20funci%5C%27on%20%7Df%5Ctext%7B%20se%20llama%20integrable%20%28seg%5C%27un%20Riemman%29%7D%5C%5C%0A%5Ctext%7Bsobre%20el%20segmento%20%7D%5Ba%2Cb%5D%5Ctext%7B%20si%20existe%20un%20n%5C%27umero%20%7DA%5Ctext%7B%20tal%20que%20para%20cualquier%7D%5C%5C%0A%5Ctext%7Bsucesi%5C%27on%20de%20particiones%20del%20segmento%20%7D%5Ba%2Cb%5D%5C%5C%20%5C%5C%0A%5Ctau_n%3D%5C%7Bx_i%5E%7B%28n%29%7D%5C%7D_%7Bi%3D0%7D%5E%7Bi%3Dk%7D%20%5C%3B%2C%5C%3B%5C%3B%5C%3Bn%3D1%2C2%2C%5Ccdots%5C%5C%20%5C%5C%0A%5Ctext%7Bpara%20la%20cual%20%7D%5Clim%5Climits_%7Bn%5Cto%20%5Cinfty%7D%5Cdelta_%7B%5Ctau_n%7D%3D0%5Ctext%7B%20y%20para%20cualquier%20elecci%5C%27on%20de%20puntos%20%7D%0A%0A%5C%5C%20%5C%5C." />

<img src="https://tex.z-dn.net/?f=%5Cxi_i%5E%7B%28n%29%7D%5Cin%20%5Cleft%5Bx_%7Bi-1%7D%5E%7B%28n%29%7D%2Cx_i%5E%7B%28n%29%7D%5Cright%5D%20%5C%3B%2C%5C%3B%20i%3D1%2C2%2C%5Ccdots%2Ck%5C%3B%5C%3B%2C%5C%3B%20n%3D1%2C2%2C%5Ccdots%5C%5C%20%5C%5C%0A%5Ctext%7Bexiste%20el%20l%5C%27imite%20de%20la%20sucesi%5C%27on%20de%20sumas%20integrales%20%7D%5C%5C%0A%5Csigma_%7B%5Ctau_n%7D%28f%2C%5Cxi_1%5E%7B%28n%29%7D%2C%5Cxi_2%5E%7B%28n%29%7D%2C%5Ccdots%2C%2C%5Cxi_%7Bk_n%7D%5E%7B%28n%29%7D%29%5C%5C%20%5C%5C%0A%5Ctext%7By%20es%20igual%20a%20%7DA%5C%5C%20%5C%5C%0A%5Cdisplaystyle%0A%5Clim%5Climits_%7Bn%5Cto%5Cinfty%7D%5Csum_%7Bi%3D1%7D%5E%7Bk_n%7Df%28%5Cxi_i%5E%7B%28n%29%7D%29%5CDelta%20x_i%5E%7B%28n%29%7D%3DA%0A%5C%5C%20%5C%5C." />

Y aquí viene la respuesta a la pregunta.

Cumpliendo se las condiciones de la definición el número A se llama integral definida de la función f sobre el segmento [a, b] <img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cint_%7Ba%7D%5Ebf%28x%29dx%3D%5Clim%5Climits_%7Bn%5Cto%5Cinfty%7D%5Csum_%7Bi%3D1%7D%5E%7Bk_n%7Df%28%5Cxi_i%5E%7B%28n%29%7D%29%5CDelta%20x_i%5E%7B%28n%29%7D" />.