Double Integration (polar)

Double Integration (polar)
Wize Concept
Double Integration (polar). If D⊂R2D \subset \mathbb{R}^2D⊂R2 is defined by
g1(θ)≤r≤g2(θ),a≤θ≤b,g_1(\theta) \le r \le g_2(\theta),\quad a\le \theta \le b,g1​(θ)≤r≤g2​(θ),a≤θ≤b,
where g1(θ)g_1(\theta)g1​(θ) and g2(θ)g_2(\theta)g2​(θ) are continuous for a≤θ≤ba \le \theta \le ba≤θ≤b and f(r,θ)f(r, \theta)f(r,θ) is continuous on DDD, then
∬Df(r,θ)dA=∫ab(∫g1(θ)g2(θ)f(r,θ)rdr)dθ\iint\limits_{D}f(r,\theta)dA = \int_a^b\Bigg(\int_{g_1(\theta)}^{g_2(\theta)}f(r,\theta)rdr\Bigg)d\thetaD∬​f(r,θ)dA=∫ab​(∫g1​(θ)g2​(θ)​f(r,θ)rdr)dθ
If D⊂R2D \subset \mathbb{R}^2D⊂R2 is defined by
h1(r)≤θ≤h2(r),c≤r≤d,h_1(r) \le \theta \le h_2(r),\quad c\le r \le d,h1​(r)≤θ≤h2​(r),c≤r≤d,

where h1(r)h_1(r)h1​(r) and h2(r)h_2(r)h2​(r) are continuous for c≤r≤dc \le r \le dc≤r≤d and f(θ,r)f(\theta, r)f(θ,r) is continuous on DDD, then
∬Df(θ,r)dA=∫cd(∫h1(r)h2(r)f(r,θ)dθ)rdr\iint\limits_{D}f(\theta,r)dA = \int_c^d\Bigg(\int_{h_1(r)}^{h_2(r)}f(r,\theta)d\theta\Bigg)rdrD∬​f(θ,r)dA=∫cd​(∫h1​(r)h2​(r)​f(r,θ)dθ)rdr

→\to→ Notes:
If the function and/or region displays some symmetry around the origin, it may be worthwhile converting from rectangular coordinates to polar.
Don't forget the extra factor of rrr in the integrand.

Practice: Double integration (polar)
Compute ∬DxdA\displaystyle\iint\limits_{D}xdAD∬​xdA, where DDD is the region bounded by x2+y2≤4x^2 + y^2 \le 4x2+y2≤4 and x≥0x \ge 0x≥0.

Extra Practice

Practice: Double Integrals in Polar Coordinates (1)

What is the volume of the region under the sphere x2+y2+z2=9, x^2+y^2+z^2=9,~x2+y2+z2=9, above the plane z=0, z=0,~z=0, and inside  x2+y2=5?~x^2+y^2=5? x2+y2=5?

Practice: Double Integrals in Polar Coordinates (2)

Find ∫D∫1+4x2+4y2dA \displaystyle\int_{D}^{}\int_{}^{}\sqrt{1+4x^2+4y^2}dA~∫D​∫​1+4x2+4y2​dA where D is the lower half of x2+y2=16x^2+y^2=16x2+y2=16

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Double Integration (polar)

Practice: Double integration (polar)

Extra Practice

Practice: Double Integrals in Polar Coordinates (1)

Practice: Double Integrals in Polar Coordinates (2)