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7.2 Sum of Area and Definite Integrals

Definite Integrals as an Accurate Estimate

  • As the number of rectangles increases, we are able to obtain more and more accurate estimations of the rectangle. Ultimately, when we split the area into infinitely many slim rectangles, the difference between the rectangle’s horizontal edge and the curve’s curvy line in the infinitely small range of x is not noticeable, and so we can get an accurate estimate.
  • In fact, we call the area under the graph of y=f(x) from x=a to x=b as the definite integral of f(x) from x=a to x=b. It is written as

\int_{a}^{b} f(x) d x

where b>a.

  • The function f is called the integrand, and a and b are the lower and upper limits of the integral.
  • Using the summation notation, it can be rewritten as
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