This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 xaxis? 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 xyplane. (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 halfstrip 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. ...
View Full
Document
 Spring '08
 FABER
 Math, Calculus

Click to edit the document details