Properties of definite integrals

1. Height of the Riemann rectangle which is y­value of f is negative. Height of the Riemann rectangle which is y­value of f is positive.
The product of positive(dx) and negative(f(x)) value is negative. The product of positive(dx) and positive(f(x)) value is positive.
Therefore, the value of the first integral is negative. Therefore, the value of the first integral is positive.
The integral∫f(x)dx is:
positive if f(x) is positive for all x­value in the interval [a,b]
negative if f(x) is negative for all x­value in the interval [a,b]
provided a < b
since the rectangle is the base times the height and the height of the rectangles are positive
and because the b­value is smaller than the a­value the change in x is negative and a negative
base times a positive height equals a negative product (area)
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2. Symmetric over the y­axis
The first integral is twice the second due the fact that it is an even function (meaning the
two sides are equal) and our regions are equidistant from the y­axis. The first contains both
sides of the function while the second is only half, or one side of the function.
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3. Symmetrical over the origin
The answer is zero because the area ­4 to 0 is negative and the area 0 to 4 is positive but they
are the same value because it is an odd function so they subtract to zero.
If f(x) is an even function then: If f(x) is an odd function then:
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4. see graph
Each y­value of h(x) is doubled to build the graph of 2h(x).
Enter h(x) into Y1.
2Y1(6) ­ 2Y1(1) = 54
see graph
54 is double 27.
5(27) = 135
The integral of a constant is equal to the constant times the
times a function integral of the function
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5. 5

6. 6