<p>Plot</p>

07f8d54f-b591-4fc0-ab84-bf7bb62aa822 223 3 5 decimal

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# # f u v pRegion M 0 : n 0 : 12 mat r 1 u length range c 1 v length range u r el v c el f 1 ≡ n n 1 + : M n 1 el u r el : M n 2 el v c el : 13 mat continue if for for M eval 31 line :

a b N pR d b a - N 1 - / : R 0 : 12 mat n 1 N range R n el a n 1 - (d * + : for R 31 line :

x Φ 1 x sign + 2 / :

y x atan2 y x atan 2 π * y - Φ * + 11 line :

www.math.ucla.edu/~ronmiech/Calculus_Problems/32B/chap13/section4/856d1/856_1.html

Integral in polar coordinates Alvaro

x y ρ φ Clear 1

car2 x y 21 mat :

N 40 :

ρφ

Ψ

v h r ρφ a 15 mat ρ φ 21 mat 1 ρ 21 mat ρ φ cos * ρ φ sin * 21 mat x 2^y 2^+ sqrt y x atan2 21 mat 0 ρ ≤ (ρ ∞ < (& 0 φ ≤ (φ 2 π * < (& 21 mat 15 mat :

dS.k h.i h.j * dq.i * dq.j * ≡

then

f S R ds int h 1 el h 2 el * f * v 1 el c d int v 2 el a b int ≡

Integrate f over R, where R is the disk with center the origin and radius 5

R φ 02 π * lelt ρ 05 lelt & :

x y R R v ρφ ≡ vectorize at : x y R 6 - 6 N pR 6 - 6 N pR pRegion augment 2 1 line

Ψ x :

f Ψ car2 r ≡ vectorize at : ρ φ cos *

I h 1 el h 2 el * f * ρ 05 int φ 02 π * int : 1.8492 1014 -^*

Integrate f over R, where R is a region in the first quadrant that lies between the circles C1 and C2.

C.1 x 2^y 2^+ 4 - :

C.1 car2 r ≡ vectorize at 4 - ρ 2^+

C.2 x 2^y 2^+ 25 - :

C.2 car2 r ≡ vectorize at 25 - ρ 2^+

R φ 0 π 2 / lelt ρ 25 lelt & :

x y R R v ρφ ≡ vectorize at : x y R 0 5 100 pR 0 5 N pR pRegion 2 1 line 2 1 sys

Ψ x y * :

f Ψ car2 r ≡ vectorize at : ρ 2^φ cos * φ sin *

I h 1 el h 2 el * f * ρ 25 int φ 0 π 2 / int : 76.125

Integrate f over R, where R is a region that lies inside the cardiod C1 and outside the circle C2.

C.1 ρ 1 - φ sin - : C.2 ρ 1 - : 21 sys

R φ 02 π * lelt ρ 11 φ sin + lelt & :

x y R R v ρφ ≡ vectorize at : x y R 2 - 2 N pR 0 2 N pR pRegion 2 1 line

Ψ 1 x 2^y 2^+ sqrt / :

f Ψ car2 r ≡ vectorize at : 1 ρ 2^sqrt /

I h 1 el h 2 el * f * ρ 11 φ sin + int φ 0 π int : 2

Integrate f over R, where R is the region bounded by the spirals C1 and C2, with 0<φ<2π.

C.1 ρ φ - : C.2 ρ 2 φ * - : 21 sys

R φ 02 π * lelt ρ φ 2 φ * lelt & :

x y R R v ρφ ≡ vectorize at : x y R 10 - 14 N pR 10 - 4 N pR pRegion 2 1 line

Ψ x 2^y 2^+ :

f Ψ car2 r ≡ vectorize at : ρ 2^

I h 1 el h 2 el * f * ρ φ 2 φ * int φ 02 π * int : 7344.4725

24 π 5^* 7344.4724

Find the area of one loop of the rose C.

C ρ 3 φ * cos - :

R φ 0 π 6 / lelt ρ 03 φ * cos lelt & :

x y R R v ρφ ≡ vectorize at : x y R 0 1 N pR 0.3 - 0.3 N pR pRegion 2 1 line

Ψ 1 :

f Ψ car2 r ≡ vectorize at : 1

I 2 h 1 el h 2 el * f * ρ 03 φ * cos int φ 0 π 6 / int * : 0.2618

π 12 / 0.2618

Alvaro