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22. Graph the region under y 5 r for ˇ 2 w # r # 5 ˇ 2 w . E 5 ˇ 2 w ˇ 2 w r dr 5 } 1 2 } (5 ˇ 2 w 2 ˇ 2 w )( ˇ 2 w 1 5 ˇ 2 w ) 5 24 23. E b 0 x dx 5 } 1 2 } ( b )( b ) 5 } 1 2 } b 2 24. E b 0 4 x dx 5 } 1 2 } ( b )(4 b ) 5 2 b 2 25. E b a 2 s ds 5 } 1 2 } ( b 2 a )(2 b 1 2 a ) 5 b 2 2 a 2 26. E b a 3 t dt 5 } 1 2 } ( b 2 a )(3 b 1 3 a ) 5 } 3 2 } ( b 2 2 a 2 ) 27. E 2 a a x dx 5 } 1 2 } (2 a 2 a )(2 a 1 a ) 5 } 3 2 a 2 } 28. E ˇ 3 w a a x dx 5 } 1 2 } ( ˇ 3 w a 2 a )( ˇ 3 w a 1 a ) 5 } 1 2 } (3 a 2 2 a 2 ) 5 a 2 29. Observe that the graph of f ( x ) 5 x 3 is symmetric with respect to the origin. Hence the area above and below the x -axis is equal for 2 1 # x # 1. E 1 2 1 x 3 dx 52 (area below x -axis) 1 (area above x -axis) 5 0 30. The graph of f ( x ) 5 x 3 1 3 is three units higher than the graph of g ( x ) 5 x 3 . The extra area is (3)(1) 5 3. E 1 0 ( x 3 1 3) dx 5 } 1 4 } 1 3 5 } 1 4 3 } 31. Observe that the region under the graph of f ( x ) 5 ( x 2 2) 3 for 2 # x # 3 is just the region under the graph of g ( x ) 5 x 3 for 0 # x # 1 translated two units to the right. E 3 2 ( x 2 2) 3 dx 5 E 1 0 x 3 dx 5 } 1 4 } 32. Observe that the graph of f ( x ) 5 ) x ) 3 is symmetric with respect to the y -axis and the right half is the graph of g ( x ) 5 x 3 . E 1 2 1 ) x ) 3 dx 5 2 E 1 0 x 3 dx 5 } 1 2 } 33. Observe from the graph below that the region under the graph of f ( x ) 5 1 2 x 3 for 0 # x # 1 cuts out a region R from the square identical to the region under the graph of g ( x ) 5 x 3 for 0 # x # 1. E 1 0 (1 2 x 3 ) dx 5 1 2 E 1 0 x 3 dx 5 1 2 } 1 4 } 5 } 3 4 } 34. Observe from the graph of f ( x ) 5 ( ) x ) 2 1) 3 for 2 1 # x # 2 that there are two regions below the x -axis and one region above the axis, each of whose area is equal to the area of the region under the graph of g ( x ) 5 x 3 for 0 # x # 1. E 2 2 1 ( ) x ) 2 1) 3 dx 5 2 1 2} 1 4 } 2 1 1 } 1 4 } 2 52} 1 4 } 35. Observe that the graph of f ( x ) 5 1 } 2 x } 2 3 for 0 # x # 2 is just a horizontal stretch of the graph of g ( x ) 5 x 3 for 0 # x # 1 by a factor of 2. Thus the area under f ( x ) 5 1 } 2 x } 2 3 for 0 # x # 2 is twice the area under the graph of g ( x ) 5 x 3 for 0 # x # 1. E 2 0 1 } 2 x } 2 3 dx 5 2 E 1 0 x 3 dx 5 } 1 2 } 36. Observe that the graph of f ( x ) 5 x 3 is symmetric with respect to the origin. Hence the area above and below the x -axis is equal for 2 8 # x # 8. E 8 2 8 x 3 dx (area below x -axis) 1 (area above x -axis) 5 0 y 2 x 2 y = f ( x ) y 1 x 1 y = f ( x ) R y 10 r 10 5 2 2 Section 5.2 211

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37. Observe from the graph below that the region between the graph of f ( x ) 5 x 3 2 1 and the x -axis for 0 # x # 1 cuts out a region R from the square identical to the region under the graph of g ( x ) 5 x 3
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