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Chapter 6 | Applications of Integration
• Familiar properties of logarithms and exponents still hold in this more rigorous context.
6.8 Exponential Growth and Decay • Exponential growth and exponential decay are two of the most common applications of exponential functions. • Systems that exhibit exponential growth follow a model of the form y = y 0 e kt . • In exponential growth, the rate of growth is proportional to the quantity present. In other words, y ′ = ky . • Systems that exhibit exponential growth have a constant doubling time, which is given by (ln2)/ k . • Systems that exhibit exponential decay follow a model of the form y = y 0 e − kt . • Systems that exhibit exponential decay have a constant half-life, which is given by (ln2)/ k . 6.9 Calculus of the Hyperbolic Functions • Hyperbolic functions are defined in terms of exponential functions. • Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas. • With appropriate range restrictions, the hyperbolic functions all have inverses. • Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas. • The most common physical applications of hyperbolic functions are calculations involving catenaries.
CHAPTER 6 REVIEW EXERCISES True or False? Justify your answer with a proof or a counterexample. 435. The amount of work to pump the water out of a half- full cylinder is half the amount of work to pump the water out of the full cylinder.
441. x = y 2 and x =3 y rotated around the y -axis using the washer method 442. x =2 y 2 − y 3 , x =0, and y =0 rotated around the x -axis using cylindrical shells For the following exercises, find a. the area of the region, b. the volume of the solid when rotated around the x -axis, and c. the volume of the solid when rotated around the y -axis. Use whichever method seems most appropriate to you. 443. y = x 3 , x =0, y =0, and x =2
436. If the force is constant, the amount of work to move an object from x = a to x = b is F ( b − a ).
437. The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution. 438. If the half-life of seaborgium-266 is 360 ms, then k = ⎛ ⎝ ln(2) ⎞ ⎠ /360. For the following exercises, use the requested method to determine the volume of the solid. 439. The volume that has a base of the ellipse x 2 /4+ y 2 /9=1 and cross-sections of an equilateral triangle perpendicular to the y -axis. Use the method of slicing. 440. y = x 2 − x , from x =1 to x =4, rotated around the y -axis using the washer method
444. y = x 2 − x and x =0
445. [T] y = ln( x )+2and y = x
446. y = x 2 and y = x
447. y =5+ x , y = x 2 , x =0, and x =1
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