M408C: The Definite Integral and the FundamentalTheorem of CalculusOctober 23, 2008Definition 1 (Definite Integral). Let f be a function on [a, b], and subdivide [a, b] into n subin-tervals each of size ∆x = (b − a)/n. Let x0= a, x1, x2, . . . , xn−1, xn= b be the endpoints of thesesubintervals, and let x∗1, x∗2, . . . , x∗nbe any sample points in these subintervals so that x∗i∈ [xi−1, xi].Then the definite integral of f from a to b isprovided the limit exists. If it does exist, we say f is integrable on [a, b].The sum on the right is called a Riemann sum. It turns out lots of functions are integrable:Theorem 1. If f is continuous on [a, b], or if f has only finitely many jump discontinuities, thenf is integrable on [a, b].In order to do some of the homework problems in 5.2, you will need the following facts about sums:nXi=1i =n(n + 1)2nXi=1i2=n(n + 1)(2n + 1)6nXi=1i3=n(n + 1)22nXi=1c = ncnXi=1cai= cnXi=1ainXi=1(ai+ bi) =nXi=1ai+nXi=1binXi=1(ai− bi) =nXi=1ai−nXi=1biYou should also know the following properties about integrals:Zbaf(x)dx = −Zabf(x)dxZaaf(x)dx = 0Zbac dx = c(b−a)Zbacf(x)dx = cZbaf(x)dxZba[f(x) + g(x)] dx =Zbaf(x)dx +Zbag(x)dxZba[f(x) − g(x)] dx =Zbaf(x)dx −Zbag(x)dxZcaf(x)dx +Zcbf(x)dx =Zbaf(x)dx f(x) ≥ 0 for x ∈ [a, b] ⇒Zbaf(x)dx ≥ 0f(x) ≥ g(x) for x ∈ [a, b] ⇒Zbaf(x)dx ≥Zbag(x)dxm ≤ f(x) ≤ M for x ∈ [a, b] ⇒ m(b − a) ≤Zbaf(x)dx ≤ M(b − a)Theorem 2 (Fundamental Theorem of Calculus). Suppose f is continuous on [a, b]. Then1. g(x) :=Zxaf(t)dt is continuous on [a, b], differentiable on (a, b) and g0(x) = f(x).2.Zbaf(x)dx = F (b) − F (a) for any antiderivative F of f, i.e. for any F0= f.11.2.3. (5.3.24) Evaluate the integralR813√x dx.Solution: We apply the Fundamental Theorem of Calculus: let f(x) =3√x, so the antiderivativeF of f is F (x) =34x4/3. Then by the FTC,Z813√x dx = F (8) − F (1) =34· (8)43−34· (1)43=34· (23)43−34· 1 =34· 24−34=34· 16
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