Post on 25-Jun-2015
description
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Height of the Riemann rectangle which is yvalue of f is negative. The product of positive(dx) and negative(f(x)) value is negative. Therefore, the value of the first integral is negative.
Height of the Riemann rectangle which is yvalue of f is positive. The product of positive(dx) and positive(f(x)) value is positive. 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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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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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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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
to the constant times the integral of the function
The integral of a constant times a function
is equal
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