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1、MIT OpenCourseWare 18.01 Single Variable CalculusFall 2006For information about citing these materials or our Terms of Use, visit: /terms.Lecture 1818.01 Fall2006Lecture 18: Denite IntegralsIntegrals are used to calculate cumulative totals, averages, areas.Area unde

2、r a curve: (See Figure 1.)1. Divide region into rectangles2. Add up area of rectangles3. Take limit as rectangles become thinabab(i)(ii)Figure 1: (i) Area under a curve; (ii) sum of areas under rectanglesExample 1. f(x) = x2, a = 0, b arbitrary1. Divide into n intervalsLength b/n = base of rectangle

3、2. Heights: 2bb n1st: x =,height =n 22b2b n2nd:x =,height =nSum of areas ofrectangles: 2 2 2 2b3bbb2bb3b nbnb=(12 + 22 + 32 + + n2)+ +n3nnnnnnn1Lecture 1818.01 Fall2006a=0bFigure 2: Area under f (x) = x2 above 0, b.We will now estimate the sum using some 3-dimensional geometry.Consider the staircase

4、 pyramid as pictured in Figure 3.n = 4nFigure 3: Staircase pyramid: left(top view) and right (side view)1st level: n n bottom, represents volume n2.2nd level: (n 1) (n 1), represents volumne (n 1)2), etc.Hence, the total volume of the staircase pyramid is n2 + (n 1)2 + + 1.Next, the volume of the py

5、ramid is greater than the volume of the inner prism:12+ 2 2+ + n2 1 (base)(height) = 1 n2 n = 1 n33and less than the volume of the outer prism:3312+ 22+ + n2 1 (n + 1)2( n + 1) = 1 (n + 1)3332Lecture 1818.01 Fall2006In all,1n312 + 22 + + n21 (n + 1)3133=333nn3nTherefore,b12222lim (1 + 2 + 3n n3+ n)

6、=b3,3and the area under x2 from 0 to b is b3 .3Example 2. f(x) = x; area under x above 0, b. Reasoning similar to Example 1, but easier, gives a sum of areas:bn2This is the area of the triangle in Figure 4.(1 + 2 + 3 + )12b(as n ) n 2bbFigure 4: Area under f (x) = x above 0,b.Pattern: 3 db= b2db3 2

7、db= bdb2The area A(b) under f (x) should satisfy A(b) = f (b).3Lecture 1818.01 Fall2006General Picturey=f(x)ciabFigure 5: One rectangle from a Riemann Sumb a Divide into n equal pieces of length = x =nPick any ci in the interval; use f (ci) as the height of the rectangle Sum of areas:f (c1)x + f (c2

8、)x + + f (cn)xnIn summation notation:f (ci )x called a Riemann sum.i=1Denition: bf (ci)x =f(x)dx called a denite integralanlimni=1This denite integral represents the area under the curve y = f(x) above a, b.Example 3. (Integrals applied to quantity besides area.) Student borrows from parents.P = pri

9、ncipal in dollars, t = time in years, r = interest rate (e.g., 6 % is r = 0.06/year). After time t, you owe P (1 + rt) = P + PrtThe integral can be used to represent the total amount borrowed as follows. Consider a function f (t), the “borrowing function” in dollars per year. For instance, if you bo

10、rrow $ 1000 /month, then f (t) = 12, 000/year. Allow f to vary overtime.Say t = 1/12 year = 1 month.ti = i/12 i = 1, , 12.4Lecture 1818.01 Fall2006f(ti) is the borrowing rate during the ith month so the amount borrowed is f(ti)t. The total is12f (ti)t.i=1In the limit as t 0, we have 10f (t)dtwhich represents the total borrowed in one year in dollars per year.The integral can also be used to represent the total amount owed. The amount owed de