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9.1 Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

  • If f is a continuous on an interval [a,b]], then the fundamental theorem of calculus states that

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

where F^{\prime}(x)=f(x), or F(x) is any antiderivative of f(x).

Note: This is an informal treatment of the fundamental theorem of calculus. Point 3 is the more formal definition of it.

  • Sometimes, it is also written as
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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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7.1 Estimating Area as Sum of Rectangles

Estimating Area as Sum of Rectangles

  • The area under a graph within an interval x \in [a,\ b] can be estimated by a sum of rectangles for smaller intervals of x.
  • There are two types of constructing these rectangles, namely the left-endpoint estimate and right-endpoint estimate. Both estimates will probably lead to a different answer.
  • There is no rule is left-endpoint estimate will yield a smaller or bigger area. It depends on the curve, but a general rule can be applied:
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