Summer 2016 Final.pdf - MAT1322-3X Solution to Final...

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MAT1322-3X Solution to Final Examination (A) Summer 2016 Solution to Final Examination (A) MAT1322-3X, Summer 2016 Part I. Multiple-choice Questions (3 u 10 = 30 marks) CECAE DCDAB 1. The area of the region under the graph of y = − 2 x 2 + 9 and above the graph of y = x 2 6 x is (A) 22; (B) 27; (C) 32; (D) 37; Solution . (C) The intersections of these curves are found by the equation 2 x 2 + 9 = x 2 6 x , 3 6 x 9 = 0, or x 2 2 x 3 = 0. Then x = − 1, x A = 3 3 3 2 2 2 3 1 1 1 ( 2 9 6 ) ( 3 6 9) 3 9 32 x x x x dx x x dx x x x ± ± ± ª º ± ² ± ² ± ² ² ± ² ² ¬ ¼ ³ ³ (E) 42. x 2 = 3. 2
1 3. Let R be the region between the graph y = 2 x and the x- axis, 0 d x d 1. A solid has R base, and the cross sections perpendicular to the x -axis are squares. The solid is shown in the following figure: as its
2. Let R be the region above the parabola y = x 2 and under the line y = 2 x . Solid B is obtained by revolving R about the y -axis . Then the volume of B is calculated by the integral 2 2 2 2 x x
MAT1322-3X Solution to Final Examination (A) Summer 2016 The volume of the solid is (E) 3. 2 X Y Z (1, 0, 0) 2 y x R (1, 2, 0)
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4. A container has the shape of an inverted circular cone as in the following figure: 2 6 4 2 6
MAT1322-3X Solution to Final Examination (A) Summer 2016 layer to a point 2 meters above the top of the container is dW = U g (3 x / 4) 2 (6 ± x ) S dx . The bottom layer has x = 0, and the top layer has x = 4. The total work needed is W = 4 2 0 9 (6 ) 16 g x x dx U S ± ³ . 5. Consider improper integral ² ³ . Which one of the following argument is true? 2 1 x dx x x f ±
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