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lecture16 - MATH 100 Lecture 16 Double Integral in polar...

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2006 Fall MATH 100 Lecture 8 1 MATH 100 Lecture 16 Double Integral in polar coordinates {} 12 Class 16. Double Integral in polar coordinates: Simple polar regions (, ) () () , ,0 2 Problem: find the volume of the solid lies b/w ( , ) with ( , ) 0 Rr r r r R& z f r fr θ θθ α β π =≤ =
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2006 Fall MATH 100 Lecture 8 2 MATH 100 Lecture 16 Double Integral in polar coordinates 12 . ** * Step 1: Cover with circular arcs and rays emanting from origin, retain the polar rectangules insider , and denote by , ,..., Step 2:Choose ( , ) in th block, form a box of volume ( , n kk k R R AA A rk fr θ ΔΔ Δ * 1 ), and the partial Riemann sum. ( , ) n n k A A = Δ Δ
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2006 Fall MATH 100 Lecture 8 3 MATH 100 Lecture 16 Double Integral in polar coordinates ** 1 1 Step 3: Refine cutting & take limit lim ( , ) Denote the limit to be ( , ) lim ( , ) which is called double polar integral. Remark n kk k n k n k n k R Vf r A f rd A f rA θ θθ →∞ = →∞ = = Δ ∫∫ s: (i) ( , ) can be or -, (ii) area of 1 R fr Rd A + =⋅
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2006 Fall MATH 100 Lecture 8 4 MATH 100 Lecture 16 Double Integral in polar coordinates 2 1 () 2 Evaluation Theorem 1 If R is a simple polar region, then ( , ) ( , ) Explanation: Area of a polar sector of angle and radius : 2 r r R fr dA fr r d rd r r βθ αθ θθ θ π =⋅ ∫∫ ∫ ∫ 2 22 nn n 1 2 Let be mean value of in the
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lecture16 - MATH 100 Lecture 16 Double Integral in polar...

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