Exercises7 - sin cos r and r θ = = b) Find the area of the...

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Exercises - 7 1) Let 2 2 {( , ) : 0, 0, 3} D x y x y x y = + , 2 ( , ) f x y xy = . a) Sketch the region D. b) Using iterated integrals in cartesian coordinates, find the area of D. c) Using iterated integrals in polar coordinates, find the area of D. d) Find absolute maximum and absolute minimum of f on D. 2) The same question, where D is the closed triangular region with vertices (1,0),(5,0), and (1,4) & f(x,y)=3+xy-x-2y. 3) Evaluate: a) 2 3 9 2 0 cos( ) y y x dx dy ∫ ∫ b) 1 2 0 0 ( ) y xy y dx dy + ∫ ∫ 4) Find volume of the solid: a) bounded by the planes z = x , y = x, x + y = 2 , and z = 0. b) under the surface z = 2x + y 2 and above the region bounded by x = y 2 and x = y 3 . c) enclosed by the paraboloid z = x 2 + 3y 2 and the planes x = 0 , y = 1 , y = x , z= 0. d) enclosed by the parabolic cylinder y = x 2 and the planes z = 3y & z = 2 + y. 5) a) Sketch the regions
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Unformatted text preview: sin cos r and r θ = = b) Find the area of the region within both of the circles sin cos r and r = = . 6) Let D be the region bounded by the circles x 2 + y 2 = 1 and x 2 + y 2 = 9. Write the area of D as iterated integrals ( do not evaluate ) a) using cartesian coordinates in two different ways b) using polar coordinates. 7) Use polar coordinates to evaluate the volume of the solid above the cone 2 2 z x y = + and below the unit sphere. 8) Find the average value of f(x,y)=(x 2 +y 2 ) 2 on the circular region with radius 2 and center (0,0). 9) Use polar coordinates to evaluate 2 2 3 9 3 2 9 ( ) x x x xy dy dx---+ ∫ ∫ 10) Let 0. a > Evaluate D x dA ∫∫ where 2 2 2 {( , ) : } D x y x y a = + ≤ . 11) Evaluate / 2 / 2 sin y x dy dx x π ∫ ∫...
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