please calculate me 1, 2, 3, 4, 5. Please write clearly for me. Thanks.
14 Tue 15 Thu Forum Grapher Calculus II : 15 Improper Integrals
The lesson is long and should be split into two or three blocks. Note that the last minute of this long video lost the sound. In the last minute I just define a indefinite integral on an interval [a, b] in the case if f is not continuous at a. This definition is also included in the notes below. Live Notes Expand All PDF-beta 1 Types of improper integrals In the past all definite integrals represented signed areas of regions that were bounded in some circle (finite regions). This lectures deals with integrals that represent areas of unbounded regions (infinite regions). We extend the definition of the definite integral to two cases: (1) if the interval is unbounded, (II) if the interval is bounded but the integrand has an infinite discontinuity. 2 I.Integrals over semi-infinite invervals When illustrating how to calculate work needed to take an object from the surface of the Earth arbitrary far away, we dealt with an integral of this form: da We showed that this integral is bounded no matter how large R is and that lime-infty for da exists. T This motivates us to define: I s(a)de = Lim f( )di . If the limit on the right is finite we say that the integral fo f(x)dx converges. Otherwise we say that fo f(x) da diverges. Similarily, If the limit on the right is finite we say that the integral fo f(x)da converges. Otherwise we say that S& f(x) dx diverges. 2 3 I. Integrals over the real line D We define in case both integrals on the right are convergent. Note that according to the above definition _ adr is divergent eventhough Rada = 0 for all R. 4 II. Integrals of functions with infinite discontinuities If f is continuous on (a, b] but f diverges to either infinity or negative infinity as x - at , then we define f(x)da = lim R rat [' f()de. Similarly, If f is continuous on [a, a) but f diverges to either infinity or negative infinity as x - b-, then we define [ f(x)da = lim R -+6 - J 4 Mango Homework. PDF-beta In each problem start with graphing a given function to get an idea what is going on. You can use our online grapher or google, or wolframalpha. 1. Explain in words why each of the following integrals is improper. (a) So the that. ( b) S/4 (c) Si T-2. (d) f2 In(t - 1)dt. 2. (a) Graph y = 1/th and y = 1/to.9. Select two viewing windows (a) [0, 10] x [0, 1], (b) [0, 100] x [0, 1]. (b) Evaluate S, and f10 dt (c) Evaluate f,100 ,# and f100 at (d) Find food if it exists. What does this integral represent? Find Si os if it exists. 3. Determine whether each integral is convergent or divergent. If it is convergent evaluate it. (a) foe-2adx. ( b) SITE- D). (c) Siya 4. Explain why fox lnada is improper. Evaluate it using the limit of a proper integral. 5. Calculate work needed to take an iphone 5 from the surface of the Earth to infinity. Google needed constants and parameters
