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WUSTL MATH 132 - m132_E1sF11

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MATH 132 EXAM I Solutions FALL 2011 1 Estimate the area under the graph of 0 B B from x to x using 6 rectangles of equal width whose height is given by the value of the function at the midpoint i e using M E F G H I J K L M N Solution B and midpoints are 2 25 2 75 3 25 3 75 4 25 4 75 M H 8 5 8 8 8 5 B B H B B I B B L B B M B B N B B 2 Which of the following integrals 3 represented by E B B F B B G J B B K B B lim Solution B 8 B 8 0 B B B B G 3 Find the average value of the function 0 B 38 B on the interval 1 E F G H I J K L M N Solution 0 1 38 B B 1 9 B l 1 1 1 4 B B lim V8 where V8 0 5 B is the Riemann sum with 8 8 5 5 being the right endpoint Using the formula 8 8 8 E 1 8 8 8 8 F 64 G 64 8 6 8 3 8 8 8 320 8 8 8 320 I 32 J K 3 8 3 8 6 8 8 8 8 8 8 M N 8 8 Solution B 8 5 5 8 8 5 8 8 5 8 8 3 8 8 8 8 8 8 8 8 0 5 5 8 8 8 8 5 8 5 8 8 8 8 8 8 8 we have R8 H 36 8 8 8 8 8 8 8 L 8 V8 0 5 B G 8 5 132 f1 sol 1 2 5 The indefinite integral 38 B 38 B B J B G where J B E 38 B F 9 B G 38 B H 9 B I 8 B J 9 B K 8 B L B M B N B 8 B 38 B Solution 9 B B 9 B 38 B B G 9 B G B G L 16 6 Evaluate the definite integral 1 E F G H I J K L M N Solution l F 3B 7 If 2 B 0 E F G then find 2w H I J K L M N w Solution 2w B B B 2 H 8 Using the substitution write the integral as an integral in the variable u A F G H I J K L M N Solution L 9 Evaluate the integral 1 0 9 sin E 1 F 1 G 1 H 1 I 1 J 1 K 1 L 1 M 1 N Solution 38 9 8 l 1 K 1 10 Evaluate E F 132 f1 sol 1 9 B l 9 B l B G H I J K L M N Solution 9 B l 9 B 9 B 11 Evaluate the integral B 1 1 B 1 Therefore 1 9 B l 9 B l B 1 9 B B 38 B l1 G 1 to 4 decimal places E 0 2576 F 0 3896 G 0 4765 H 0 5143 I 0 5896 J 0 6854 K 0 7453 L 0 79 M 0 8642 N 0 9468 Solution 1 1 Next do each integral seperately 1 1 38 l 1 F l 12 Evaluate the integral A F G K L M B B B where and b are positive constants D N I J Solution B B B l H 13 Find the area of the region enclosed by the curve C B B and the line C E 8 F G H I J K L M N Solution C B B is a parabola with vertex at 1 1 opening downward and it intersects the line C at 1 3 and E B B B B B B B B B l L f sol 4 14 Find the area of the region enclosed by the curve B C and B C E 8 F G H I J K L M N Solution B C is a parabola with vertex at 2 0 opening to the left The other parabola B C has vertex at the origin and also opens to the left The intersection points are at 1 1 and 1 1 So as functions of C 0 C C 8 1 C C So the area is given by C C C C C E Find the volume of the solid generated by revolving the region bounded by the curve C B and the lines C B about the line B E 1 F 1 G 1 H 1 I 1 J 1 K 1 L 1 M 1 N 1 Solution The region can be given as 0 x 1 0 y x The line segments parallel to the line of rotation x 2 are given by some x 0 x 1 Using the Shell Method we get the volume as Z 1 B B B 1 B B B 1 N 16 Find the volume of the solid generated by revolving the region bounded by the curve C B and the lines B C about the C B3 E 1 F 1 G 1 H 1 I 1 J 1 K 1 L 1 M 1 N 1 Solution The region can be given as 0 y 2 0 x y2 The line segments perpendicular to the line of rotation y axis are given by some y 0 y 2 Using the Washer Method we get the volume as Z 1 C C 1 C C 1 K f1 sol 1 5 PART II CLEARLY WRITE YOUR SOLUTION AND HOW YOU GOT IT 17 Using substitution find the general solution for the following two indefinite integrals 9 B B 38 B B B Solution 38 B B B G 38 B G 9 B B B B B Solution B B B B B B B G B B B B G 1 Consider the solid obtained by rotating the region bounded by B C B and C about the x axis a Find the volume of the above solid using the washer method Solution For the Washer Method we can describe the reion as 0 B 8 B y Z 1 B B 1 B B 1 B B l 1 b Now find the volume of the above solid using the method of cylindrical shells Solution For the Shell Method we should describe the region as 0 C B C Z 1 C C C 1 C C 1 C l 1


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WUSTL MATH 132 - m132_E1sF11

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