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13.2 DoubleIntegrals

13.2 DoubleIntegrals - 1 3.2  D o u b l e  I n t e g r...

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Unformatted text preview: 1 3 .2  D o u b l e  I n t e g r a l s  o v e r  G e n e r a l  R e g io n s   Suppose the region is not rectangular.  Integrate  π 2 cosθ 0 0 ∫∫ e sin θ drdθ                         Def:  An xy region (or xy plane) is called a type I region if any  vertical strip (in the y direction) always has the same upper  and lower boundaries and the region can be described by a set  of inequalities  a ≤ x ≤ b and h(x) ≤ y ≤ g(x).      gHxL hx a   b   Def:  An xy region (or xy plane) is called a type II region if any  horizontal strip (in the x direction) always has the same left  and right boundaries and the region can be described by a set  of inequalities  c ≤ y ≤ d and h(y) ≤ x ≤ g(y).    c hHyL gHyL d       b g(x ) ∫∫ If type II use   ∫ ∫ If type I  use  f ( x, y ) dydx   a h(x ) d g( y ) c h(y ) f ( x, y ) dxdy       Ex.  Suppose R is bounded by y = x2 and x + y = 6.                                Suppose R is the triangular region bounded by (1, 3), (2, 1)   and (4, 4).                                            Sketch the regions of integration and reverse the order of  integration.    63 Ex.                ∫ ∫ f ( x, y ) 01 4 Ex.                  0 0 e Ex.                y ln x ∫∫ ∫∫ 1 0 dydx   f ( x, y ) dxdy   e y dydx   1 Ex.                    ∫∫ y +1 −1 − y +1 f ( x, y ) dxdy   π Ex.  Integrate                                    π 0 x ∫∫ sin y dydx   y Do: Sketch the region and reverse the order of integration for  1 2.                                                  ∫∫ 2 1.     y y2 0 y2 ∫∫ 1 y f ( x, y ) dxdy   f ( x, y ) dxdy   Ex.  Find the volume of the solid  whose base is the region in  the xy‐plane that is bounded by the parabola y = 4 – x2 and the  line y = 3x, while the top of the solid is bounded by the plane   z = x + 4.   ...
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