# This is not easy to prove in general but for the

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This is not easy to prove in general, but for the Property 5 can be seen from the geometric interp from a to c plus the area from c to b is eq y f x y c a f x dx y b c f x dx 4 3 y 1 0 4 3 x 2 dx y 1 0 4 dx y 1 0 x y 1 0 4 dx 4 1 0 y 1 0 4 3 x 2 dx y 1 0 4 dx y 1 0 3 x 2 d f t f t y b a f x dx lim n l n i 1 f x i lim n l n i 1 f x y b a f x t x dx lim n l n i 1 f x f t 374 |||| CHAPTER 5 INTEGRALS y 0 x a b f g f+g FIGURE 14 j   [ ƒ+© ]   dx= j   ƒ   dx+ j   ©   dx a b a b a b N Property 3 seems intuitively reasonable because we know that multiplying a function by a positive number stretches or shrinks its graph vertically by a factor of . So it stretches or shrinks each approximating rectangle by a factor and therefore it has the effect of multiplying the area by . c c c c FIGURE 15 0 y x a b c y=ƒ It is well-known that ˆ 1 0 e x 2 dx does not have a closed-form expression. However, one can estimate its value from above using Property (5): for any x in [ 0, 1 ] , we have x 2 x and so e x 2 e x . Therefore, ˆ 1 0 e x 2 dx ˆ 1 0 e x dx = e - 1.
96 Integration 5.3 Fundamental Theorem of Calculus In the previous sections, we talked about the indefinite and definite integrals of a given function f ( x ) . The former is merely the anti-derivative of f ( x ) , while the latter represents the area under the graph y = f ( x ) . They are a priori different things, but the following important theorem (Fundamental Theorem of Calculus) relates them together. Essentially, this theorem asserts that finding area under the graph of a continuous function can be done by computing anti-derivative. This explains why both anti-derivatives and area under a graph are called integrals , and that they use similar symbols. Theorem 5.4 — Fundamental Theorem of Calculus. Given a function f ( x ) which is continu- ous on [ a , b ] , and F ( x ) is an anti-derivative of f ( x ) , i.e. F 0 ( x ) = f ( x ) , then we have: d dx ˆ x a f ( t ) dt = f ( x ) and ˆ b a f ( x ) dx = F ( b ) - F ( a ) i Note that the definite integral ˆ x a f ( t ) dt is a function of x but not t . Graphically, this number represents the area under the graph y = f ( t ) from t = a to t = x as shown below: Isaac Barrow (1630–1677), discovered that thes related. In fact, he realized that differentiation and Fundamental Theorem of Calculus gives the pre derivative and the integral. It was Newton and Leib used it to develop calculus into a systematic mathe that the Fundamental Theorem enabled them to c without having to compute them as limits of sums The first part of the Fundamental Theorem deals of the form where is a continuous function on and v depends only on , which appears as the variable u number, then the integral is a definite num also varies and defines a function of den If happens to be a positive function, then graph of from to , where can vary from to tion; see Figure 1.) EXAMPLE 1 If is the function whose graph is , find the values of , , , rough graph of .