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Definite Integrals

11.1 Finding Area Under a Curve

Finding Area Under a Curve

  • Integration, or in exact definite integrals, are useful in finding areas under a graph.
  • In this course, the total area is more of a concern compared to the signed area, if the questions requests you to find an area of a region. Therefore, it can be found using definite integrals as above, but be mindful of whether the area is above the x-axis, below, or mixed.
  • In the following, consider the function f(x) and the area A of the region contained by the curve, the axis and the range x\in[a,b].

i) If f(x)\geq0 for all x\in[a,b], then we have

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

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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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Integration

This tutorial covers material encountered in chapter 11 of the VCE Mathematical Methods Textbook, namely: Integration If you’re thinking “that’s all?” do not be deceived.… Read More »Integration