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Extra resources for Calculus 2c-5, Examples of Plane Integrals

Example text

D There are at least three different solutions: 1) Reduction in rectangular coordinates. 2) Reduction in polar coordinates. 3) Reduction in a translated polar coordinate system. 5 2 Figure 37: The quarter disc B. com 50 Calculus 2c-5 Plane integrals, polar coordinates I First method. Reduction in rectangular coordinates. The set B is described in rectangular coordinates by 2x − x2 , x ∈ [1, 2]}. B = {(x, y) | 0 ≤ y ≤ Hence 2 yx dS √ 2 2x−x2 = 1 B yx2 dy dx = 0 x4 x5 − 2 5 1 2 = 2 = 1 1 2 2 1 x2 {2x − x2 } dx = 1 2 2 2x3 − x4 dx 1 1 1 75 − 62 13 {16 − 1} − {32 − 1} = = .

7, when a = 1 and b = 1 . 2e 7) The set is an annulus shaped domain which is neither nice in a rectangular description nor in a polar description. com 34 Calculus 2c-5 Plane integrals, polar coordinates When we reduce the plane integral it is fairly simple to get π x B 3 (x2 + y 2 ) 2 π = −π 1 cos ϕ dS = −π 3 b exp(a cos ϕ) π cos ϕ{− ln b − a cos ϕ}dϕ = −a π · d −π dϕ = −π cos ϕ · [ln ]1b exp(a cos ϕ) dϕ cos2 ϕ dϕ = −aπ. 8, when a = 1 and b = 2. 7. 7. 9. 9) The set B is a circular sector os shown on the figure.

3) By the usual reduction the area is here computed in the following way, dS B = π 3 −π 3 1 2 a − 4a cos ϕ cos ϕ π 2 dϕ = = a2 [tan t + 4 sin 2ϕ]03 = a2 tan a2 ·2 2 π 3 0 2π π + 4 sin 3 3 1 − 8 + 8 + 8 cos 2ϕ dϕ cos2 ϕ √ √ √ 3 = 3 3 a2 . 3 Find the area of the plane domain B, which is bounded by (i) a part of Archimedes’s spiral given in polar coordinates by = aϕ, ϕ ∈ [0, π], and (ii) the part of the negative X-axis given by (y = 0 and x ∈ [−πa, 0]), or ∈ [0, πa]). (ϕ = π and A Area in polar coordinates.

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Calculus 2c-5, Examples of Plane Integrals by Mejlbro L.

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