105-2-AreasDefiniteIntegrals

105-2-AreasDefiniteIntegrals - 2 Areas Denite Integrals...

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2 Areas & Definite Integrals Note: There are no pictures in these notes, so often you’ll have to draw the relevant picture yourself (which shouldn’t be hard), or you’ll have to find a similar picture in the book. 2.1 Definite Integrals Temporary definition of definite integral We define the new concept of a definite integral : R b a f ( t ) dt is the net area under the graph of f ( x ) between a and b . Several specifications should be made: ’under’ means between the graph and the x -axis. ’net’ area means that we count area above the x -axis positively, and we count area below the x -axis negatively. Normally a < b and we look at the area from left to right. If b < a , then the area should be counted negatively, as if it’s going from right to left. Although we’re using the same symbol R as in the last chapter, right now these symbols have nothing to do with each other: the indefinite integral R f ( t ) dt stands for the general antiderivative of f , while R b a f ( t ) dt is defined as an area. This definition is ’temporary’ because we’re not defining what area is; it’s an intuitive notion, but in mathematics that’s not quite good enough. Later in this chapter we’ll see how to define area and definite integrals precisely, using Riemann sums. The function f ( t ) inside is referred to as the integrand . The numbers a and b are the limits of integration ; a is the lower limit and b the upper limit . Properties of definite integrals We can deduce several properties of definite integrals that we’ll need later. To begin with: Z a a f ( t ) dt = 0 , Z a b f ( t ) dt = - Z b a f ( t ) dt. The first follows from the word ’between’ – there is no area between a and a . The second captures the remark above that if you go from right to left instead of from left to right, the area counts negatively. The next two properties describe what happens to a definite integral if you modify the integrand: Z b a c · f ( t ) dt = c · Z b a f ( t ) dt, Z b a ( f ( t ) + g ( t )) dt = Z b a f ( t ) dt + Z b a g ( t ) dt. The first says that the area under a scaled function equals the area under that function, scaled by the same factor. The second says that to calculate the area under the sum of two functions you can simply take the areas under the two functions separately, and add the two results together. But note that this only makes sense if the two integrals have the same 1
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limits of integration. Both of these make perfect sense if you think of how areas work. Note that these happen to exactly match the properties of the indefinite integral R f ( t ) dt . Finally, we can combine two integrals that have the same integrand but different limits of integration, but only in the special case where the upper limit of one integral is the lower limit of the other: Z b a f ( t ) dt + Z c b f ( t ) dt = Z c a f ( t ) dt.
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Fundamental Theorem of Calculus (FTC) We have defined indefinite integrals (antiderivatives) and definite integrals (like area func- tions) in very different ways; we even used the same symbol. Of course, this was for a reason,
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