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Lecture 13-2 and 13-3

# Lecture 13-2 and 13-3 - Z-1 Z √ y 1-√ y 1 f x y dxdy...

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13.2 Double Integrals over General Regions 13.3 Area by Double Integrals In R 2 : Area between curves y = f ( x ) and y = g ( x ): OR area between curves x = f ( y ) and x = g ( y ): In R 3 : Area of a region R bounded by y = f ( x ) and y = g ( x ) is Area of a region R bounded by x = f ( y ) and x = g ( y ) is 1

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So, in general: Defn. Area The area of a closed, bounded plane region R is Defn. Volume The volume of the surface below z = f ( x, y ) and above the region R is Note When setting up integrals, outer limits are ALWAYS constants! Example 1 Suppose R is bounded by y = x 2 and x + y = 6. Find the volume under z = f ( x, y ) over R . 2
Example 2 Evaluate ZZ D y x dA where D is the region bounded by x = 2 y , x = y , y = 1 and y = 2. Finding Regions of Integration Example 1 Z 6 0 Z 3 1 f ( x, y ) dydx Example 2 Z 4 0 Z y 0 f ( x, y ) dxdy 3

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Example 3 Z π 0 Z sin x 0 ydydx Reversing Order of Integration Example 1 Z π 0 Z π x sin y y dydx . Example 2 Z e 1 Z ln x 0 e y dydx . Sketch the region and reverse the order of integration. 4
Example 3

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Unformatted text preview: Z-1 Z √ y +1-√ y +1 f ( x, y ) dxdy Defn. Average Value of f The average value of f over a region R is Example Find the average value of x cos xy over the rectangle R : 0 ≤ x ≤ π , 0 ≤ y ≤ 1. 5 DO. 1. Sketch the region of integration for Z 2 1 Z y 2 y f ( x, y ) dxdy . 2. Reverse the order of integration of Z 1 Z y 2 y f ( x, y ) dxdy . 3. SETUP only. Find the area of the region bounded by x = 1 2 y 2-3 and x-y = 1. 6 4. SETUP only. Find the volume of the solid whose base is the region in the xy-plane that is bounded by the parabola y = 4-x 2 and the line y = 3 x , while the top of the solid is bounded by the plane z = x + 4. 7...
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Lecture 13-2 and 13-3 - Z-1 Z √ y 1-√ y 1 f x y dxdy...

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