McGill0.M262.PracticeQuestions

McGill0.M262.PracticeQuestions - (9) ( Hard! ) Does the...

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SOME PRACTICE QUESTIONS ON MULTIPLE INTEGRATION MATH 262, AUTUMN 2009. (1) Compute ± S y 2 d A ( 1 + x 2 ) , where S is the infnite strip 0 < y < 1. (2) Compute ± R d A ( 5 x + 3 y + 12 ) 2 , where R is the region bounded by the triangle with vertices ( 0 , 0 ) , ( 0 , 2 ) and ( 1 , 0 ) . (3) Compute ± D ( 3 x + 6 y + 1 ) d A , where D is the circle x 2 + y 2 = 6 y . (4) Draw the region D specifed by the inequalities y ± 7 x and x ± y 2 - 4 y and then compute its area. (5) Evaluate the Following integral by interchanging the order oF inte- gration: ± 1 0 ± 2 2 x 2 x 3 sin ( 3 y 3 ) d A . (6) What is the average value oF f ( x , y )= xy over the region between the curve y = 1 / ( 1 + x 2 ) and the x -axis? What is the average value oF f over the region defned by x ± 0, y ± 0and y ² 1 / ( 1 + x 2 ) ? (7) ±ind the volume lying between the paraboloid z = 4 - x 2 - y 2 and the xy -plane. (8) ±ind the volume oF intersection oF the sphere x 2 + y 2 + z 2 = 4and the cylinder x 2 + y 2 = 2 x .
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2S O M E P R A C T I C E Q U E S T I O N S O N M U L T I P L E I N T E G R A T I O N
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Unformatted text preview: (9) ( Hard! ) Does the integral H d A 1 + x + y converge, if H is the half-strip deFned by 0 < x < 1 and y > 0? What if we replace y by y 2 in the integrand? You might like to check (and use!) the inequalities ln ( 1 + t ) t and 1 1 ( 1 + u ) 2 , which are valid for t 0 and u 0, respectively. (10) ( Hard! ) Consider the integral R x 2-y 2 x 2 + y 2 d x d y , where R is the region bounded by the curves y = x , y = x / 3, xy = 1 2 and xy = 2. Evaluate this integral using both (a) polar coordinates, r = x 2 + y 2 and = arctan ( y / x ) , and (b) the coordinates u = xy and v = y / x ....
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This note was uploaded on 10/02/2010 for the course ENGINEERIN MATH 262 taught by Professor Rix during the Fall '10 term at McGill.

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McGill0.M262.PracticeQuestions - (9) ( Hard! ) Does the...

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