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Introduction to Integrals

The Definite Integral

Problems for "The Area as a Sum"

Problems for "The Definite Integral"

The definite integral is a convenient notation used the represent the left-hand and right-hand approximations discussed in the previous section. f (x)dx means the area of the region bounded by f , the y -axis and the lines x = a and x = b. Writing f (x)dx is equivalent to writing

f(xk)Δx    

on the interval [a, b] , but it is a much more compact way of doing so. Note also the similarity between the two expressions. This should serve as a clear reminder that the definite integral is just the limit of right-hand and left-hand approximations.

Unlike the indefinite integral, which represents a function, the definite integral represents a number, and is simply the signed area under the curve of f . The area is considered "signed" because according to the method of calculating the areas by subdivisions, the regions located below the x -axis will be counted as negative, and the regions above will be counted as positive. Negative regions cancel out positive regions, and the definite integral represents the total balance between the two over the given interval. For example, find

sin(x)dx    

Based on the picture of the region being considered, it should be clear that the answer is zero. Here, the negative region is exactly the same size as the positive region:

Figure %: Graph of f (x) = sin(x) on the interval [- Π, Π]

Properties of the Definite Integral

The definite integral has certain properties that should be intuitive, given its definition as the signed area under the curve:

  1. cf (x)dx = c f (x)dx
  2. f (x)+g(x) dx = f (x)dx + g(x)dx
  3. If c is on the interval [a, b] then

    f (x)dx = f (x)dx + f (x)dx    

    This means that we can break up a graph into convenient units and find the definite integral of each section and then add the results to find the total signed area for the whole region.

    The Fundamental Theorem of Calculus

    The fundamental theorem of calculus, or "FTC", offers a quick and powerful method of evaluating definite integrals. It states: if F is an antiderivative of f , then

    f (x)dx=F(b) - F(a)    

    For example,

    x 2 dx = (1)3 - (0)3 =    

    Often, a shorthand is used that means the same as what is written above:

    x 2 dx = x 3 =    

    One interpretation of the FTC is that the area under the graph of the derivative is equal to the total change in the original function. For example, recall that velocity is the derivative of position. So,

    v(t)dt=s(b) - s(a)    

    This means that the change in area under the velocity curve represents the total change in position.

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