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6 A swimming pool is V-shaped Each end of the pool is an inverted equilateral triangle of side 8 feet The pool is 20 feet long The pool is full Set up, but do not calculate, the integral for the pressure on one end of the pool 7 A man climbs a ladder with a 80-pound sack of sand that is leaking one pound per minute If he climbs steadily at the rate of 4 feet per minute, and if the ladder is 20 feet high, then how much work does he do in climbing the ladder 8 Because of a prevailing wind, the force that opposes a certain runner is 2x 2 + 3x + 4 pounds at position x How much work does this runner perform as he runs from x = 2 to x = 10 (with distance measured in feet) 9 Set up, but do not evaluate, the integrals for each of the following arc length problems (a) The length of the curve y = cos x, 0 x /2 (b) The length of the curve x 2 = y3 , 1 x 4 (c) The length of the curve sin y = x, 0 y /2 (d) The length of the curve y = x 2 , 1 x 3 10 Set up the integral for, but do not calculate, the average value of the given function on the given interval 2 (a) f (x) = sin x [1, 4] (b) g (x) = tan x [0, /3] 2x , [ 2, 2] (c) h(x) = x+1 sin x [ /2, ] (d) f (x) = 2 + cos x 11 Write down the sum that will estimate the given integral using the method of rectangles with mesh of size k You need not actually evaluate the sum.

y, y', y", y'''

Fig 10-10

Since the velocities are equal at the point q = b1 differentiating Eqs (225) and (226) yields b C1 = C2 - 1 b1 Solving Eq (229) and (230) gives C1 = Therefore for 0 < q b, displacement y1 = y1 = y1 = h 2 q b1b 2h q b1b 2h b1b (232) h , b1b C2 = h (230)

2 0

(231)

2 4

(233)

There are a large, large number of trigonometric integrals Some are relatively easy Most are relatively difficult Solving trigonometric integrals involves changes of variables and using trigonometric identities and a good bit of ingenuity, imagination some might call it The following several problems demonstrate the more popular techniques (did someone say tricks ) for solving trigonometric integrals

(234)

12 Do each of the problems in Exercise 11 with method of rectangles replaced by trapezoid rule

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( b - q )2

1 The number 396545454 written as a rational fraction is (a) 3894/999 (b) 3152/1000 (c) 39258/9900 (d) 41445/9999 (e) 5312/2122 2 The intersection of the sets [2, 4] and (3, 6) is (a) (3, 4] (b) [2, 6) (c) (3, 5) (d) (2, 6] (e) [3, 4] 3 The intersection of the sets {(x, y) : x 2 + y2 < 1} and {(x, y) : y > 0} is (a) a half-disc with boundary (b) a quarter-disc (c) a disc without boundary (d) a disc with boundary (e) a half-disc without boundary 4 The line through the points (2, 4) and (1, 6) has slope (a) 10 (b) 10 (c) 5 (d) 3 (e) 2

(235)

(b - q )

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