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480 Chapter 8: Integration Techniques of 70. Find the volume of the solid generated by revolving the region about the x-axis. Plot the integrand (sin t)/t for t > 0. Is the sine-integral function everywhere increasing or decreasing? Do you think Si (x) = 0 for x > 0 ? Check your answers by graphing the function Si (x) for 0 < x < 25. 71. Find the area of the region that lies between the curves y = secx and y — tanx from x = 0 to x = IT/2. Explore the convergence of 72. The region in Exercise 71 is revolved about the x-axis to generate a solid. dt. a. Find the volume of the solid. b. Show that the inner and outer surfaces of the solid have infinite area. 73. Estimating the value of a convergent improper integral whose domain is infinite If it converges, what is its value? Error function The function erf (x) 74 J0 a. Show that ' dx -e^ < 0.000042, and hence that f™ e ^ dx < 0.000042. Explain why this means that /0°° e~%1 dx can be replaced by e~x~ dx without introducing an error of —— dt. J o V77 If it converges, what appears to be its value? You will see how to confirm your estimate in Section 14.4, Exercise 41. 74. The infinite paint can or Gabriel's horn As Example 3 shows, the integral Jx (dx/x) diverges. This means that the integral 77. Normal probability distribution ~dx, + rb 1 + x^dX > 2jr The function f i x ) = 277 which measures the surface area of the solid of revolution traced out by revolving the curve y = 1/x, 1 < x, about the x-axis, diverges also. By comparing the two integrals, we see that, for every finite value b > 1, xV important applications in b. Explore the convergence of lUj b. Evaluate J0 e x dx numerically. l7T dt, has a. Plot the error function for 0 < x < 25. magnitude greater than 0.000042. 277^1 2e~ V 77 called the error function, probability and statistics. is called the normal probability density function with mean /x and standard deviation cr. The number JUL tells where the distribution is centered, and a measures the "scatter" around the mean. From the theory of probability, it is known that j\ X* dX' f f ( x ) dx = 1. 1. In what follows, let \x — 0 and a Y\ a. Draw the graph of /. Find the intervals on which / is increasing, the intervals on which / is decreasing, and any local extreme values and where they occur. b. Evaluate / /(x) dx J-n However, the integral for n = 1,2, and 3. 771 — ) dx c. Give a convincing argument that for the volume of the solid converges. (Hint: Show that 0 < f ( x ) < e~x/1 for x > 1, and for b > 1, a. Calculate it. b. This solid of revolution is sometimes described as a can that does not hold enough paint to cover its own interior. Think about that for a moment. It is common sense that a finite amount of paint cannot cover an infinite surface. But if we fill the horn with paint (a finite amount), then we will have covered an infinite surface. Explain the apparent contradiction. 75. Sine-integral function 3 e^2dx->0 78. Show that if /(x) is integrable on every interval of real numbers and a and b are real numbers with a < b , then a. The integral Si(x) = f ( x ) dx and f-oo f i x ) sm t dt, called the sine-integral function, has important applications in optics. /(x) dx = 1. as b — >co.) b. dx and f ( x ) dx both converge if and only if /(x) dx both converge. f ( x ) dx + fa°° f ( x ) dx = the integrals involved converge. f i x ) dx + f i x ) dx when