# Chapter 7 35 prove the formula for the area of a

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Chapter 4 / Exercise 47
Trigonometry
McKeague/Turner
Expert Verified
CHAPTER 7 35. Prove the formula for the area of a sector of a circle with radius and central angle . [ Hint: Assume and place the center of the circle at the origin so it has the equation . Then is the sum of the area of the triangle and the area of the region in the figure.] ; 36. Evaluate the integral Graph the integrand and its indefinite integral on the same screen and check that your answer is reasonable. 37. Find the volume of the solid obtained by rotating about the -axis the region enclosed by the curves , , , and . 38. Find the volume of the solid obtained by rotating about the line the region under the curve , . 39. (a) Use trigonometric substitution to verify that (b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a). ¨ ¨ y=œ „„„„„ t 0 y a x y x 0 s a 2 t 2 dt 1 2 a 2 sin 1 x a 1 2 x s a 2 x 2 0 x 1 x 1 y x s 1 x 2 y 0 x 0 x 3 x y 9 x 2 9 y dx x 4 s x 2 2 O x y R Q ¨ P POQ PQR x 2 y 2 r 2 A 0 2 r A 1 2 r 2 40. The parabola divides the disk into two parts. Find the areas of both parts. 41. A torus is generated by rotating the circle about the -axis. Find the volume enclosed by the torus. 42. A charged rod of length produces an electric field at point given by where is the charge density per unit length on the rod and is the free space permittivity (see the figure). Evaluate the integral to determine an expression for the electric field . 43. Find the area of the crescent-shaped region (called a lune ) bounded by arcs of circles with radii and . (See the figure.) 44. A water storage tank has the shape of a cylinder with diam- eter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 ft, what percentage of the total capacity is being used? y 1 2 x 2 x 2 y 2 8 R r R r 0 x y L P  (a, b) E P 0 E P y L a a b 4 0 x 2 b 2 3 2 dx P a , b L x x 2 y R 2 r 2 In this section we show how to integrate any rational function (a ratio of polynomials) by expressing it as a sum of simpler fractions, called partial fractions, that we already know how to integrate. To illustrate the method, observe that by taking the fractions and to a common denominator we obtain If we now reverse the procedure, we see how to integrate the function on the right side of 2 x 1 1 x 2 2 x 2 x 1 x 1 x 2 x 5 x 2 x 2 1 x 2 2 x 1 7.4 Integration of Rational Functions by Partial Fractions
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Chapter 4 / Exercise 47
Trigonometry
McKeague/Turner
Expert Verified
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS 509 this equation: To see how the method of partial fractions works in general, let’s consider a rational function where and are polynomials. It’s possible to express as a sum of simpler fractions pro- vided that the degree of is less than the degree of . Such a rational function is called proper. Recall that if where , then the degree of is and we write .