calculate-0-1-2-cos-x-2-y-2-dxdy-

Question Number 41346 by maxmathsup by imad last updated on 05/Aug/18

c a l c u l a t e \int \int_{{[0, 1]}^{2}} c o s (x^{2} + y^{2}) d x d y .

Commented by alex041103 last updated on 06/Aug/18

W h a t d o e s {[0, 1]}^{2} m e a n ?

Commented by math khazana by abdo last updated on 07/Aug/18

t h a t m e a n [0, 1] \times [0, 1]

Commented by alex041103 last updated on 07/Aug/18

O K . B u t w h a t d o e s i t m e a n s a s a r e a g o n ?

I^{'} m a s k i n g w h a t i s t h e r e a g o n w h i c h

i s d e f i n e d b y {[0, 1]}^{2} o r [0, 1] \times [0, 1]

Commented by MrW3 last updated on 08/Aug/18

0 ⩽ x ⩽ 1

0 ⩽ y ⩽ 1

Commented by maxmathsup by imad last updated on 08/Aug/18

l e t A = \int \int_{{[0, 1]}^{2}} c o s (x^{2} + y^{2}) d x d y c h a n g e m e n t x = r c o s θ a n d y = r s i n θ

A = \int \int_{w} c o s (r^{2}) r d r d θ l e t f i n d w

w e h a v e 0 ⩽ x ⩽ 1 a n d 0 ⩽ y ⩽ 1 \Rightarrow 0 ⩽ x^{2} + y^{2} ⩽ 2 \Rightarrow 0 < r ⩽ \sqrt{2} a n d 0 ⩽ θ ⩽ \frac{π}{2}

A = \int_{0}^{\frac{π}{2}} (\int_{0}^{\sqrt{2}} r c o s (r^{2}) d r) d θ = \frac{π}{2} {[\frac{1}{2} s i n (r^{2})]}_{0}^{\sqrt{2}}

= \frac{π}{4} {s i n (2)} \Rightarrow A = \frac{π s i n (2)}{4} .

Commented by alex041103 last updated on 09/Aug/18

T h a t i s w r o n g . F o r a g i v e n a n g l e θ,

0 ⩽ r ⩽ s e c (θ)

s o w e h a v e :

A = \int_{0}^{π / 2} (\int_{0}^{s e c θ} r c o s (r^{2}) d r) d θ = \frac{1}{2} \int_{0}^{π / 2} s i n ({s e c}^{2} θ) d θ \dots

G o o d l u c k s o l v i n g t h a t \dots : D