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Unformatted text preview: SOME PRACTICE QUESTIONS ON MULTIPLE INTEGRATION MATH 262, AUTUMN 2009. (1) Compute y2 dA , 2 S (1 + x ) where S is the inﬁnite strip 0 < y < 1. (2) Compute dA
R , (5x + 3y + 12)2 where R is the region bounded by the triangle with vertices (0, 0), (0, 2) and (1, 0). (3) Compute (3x + 6y + 1) dA, where D is the circle = 6y. (4) Draw the region D speciﬁed by the inequalities y 7x and x and then compute its area. (5) Evaluate the following integral by interchanging the order of integration:
1 0 2 2x2 D x2 + y2 y2 − 4y x3 sin 3y3 dA. (6) What is the average value of f (x, y) = xy over the region between the curve y = 1/ 1 + x2 and the x-axis? What is the average value of f over the region deﬁned by x 0, y 0 and y 1/ 1 + x2 ? (7) Find the volume lying between the paraboloid z = 4 − x2 − y2 and the xy-plane. (8) Find the volume of intersection of the sphere x2 + y2 + z2 = 4 and the cylinder x2 + y2 = 2x. 2 OME PRACTICE QUESTIONS ON MULTIPLE INTEGRATION MATH 262, AUTUMN 2009. S (9) (Hard!) Does the integral dA H 1+x+y converge, if H is the half-strip deﬁned by 0 < x < 1 and y > 0? What if we replace y by y2 in the integrand? You might like to check (and use!) the inequalities 1 , 0 ln (1 + t ) t and 1 (1 + u)2 which are valid for t 0 and u 0, respectively. (10) (Hard!) Consider the integral x2 − y2 dx dy, 2 2 R x +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 = x2 + y2 and θ = arctan (y/x), and (b) the coordinates u = xy and v = y/x. ...
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