hi-calculate-A-x-2-y-2-dxdy-with-A-x-2-a-2-y-2-b-2-1-

Question Number 138257 by greg_ed last updated on 11/Apr/21

\boldsymbol hi!

\boldsymbol calculate :

\int \int_{A} (x^{2} - y^{2}) d x d y w i t h A = {\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} ⩽ 1}

Answered by mathmax by abdo last updated on 11/Apr/21

we consider the diffeomorphism {\begin{cases} x = arcos θ \\ y = brsin θ \end{cases}

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = r^{2} (\cos^{2} θ + \sin^{2} θ) = r^{2} ⩽ 1 \Rightarrow o ⩽ r ⩽ 1

\int \int (x^{2} - y^{2}) dxdy = \int_{0 ⩽ r < 1} \int_{0}^{2 π} (a^{2} r^{2} \cos^{2} θ - b^{2} r^{2} \sin^{2} θ) abrdrd θ

= ab \int_{0}^{1} r^{3} dr \int_{0}^{2 π} (a^{2} \cos^{2} θ - b^{2} \sin^{2} θ) d θ but

\int_{0}^{1} r^{3} dr = {[\frac{r^{4}}{4}]}_{0}^{1} = \frac{1}{4}

\int_{0}^{2 π} (a^{2} \cos^{2} θ - b^{2} \sin^{2} θ) d θ = \frac{a^{2}}{2} \int_{0}^{2 π} (1 + \cos (2 θ)) d θ

- \frac{b^{2}}{2} \int_{0}^{2 π} (1 - \cos (2 θ)) d θ = π a^{2} + \frac{a^{2}}{4} {[\sin (2 θ)]}_{0}^{2 π}

- π b^{2} + \frac{b^{2}}{2} {[\sin (2 θ)]}_{0}^{2 π} = π a^{2} + 0 - π b^{2} + 0 = π (a^{2} - b^{2}) \Rightarrow

\int \int_{A} (x^{2} - y^{2}) dxdy = \frac{ab}{4} . π (a^{2} - b^{2}) = \frac{π}{4} (a^{3} b - {ab}^{3})

Commented by greg_ed last updated on 11/Apr/21

thank you very much, dear friend mathmax by abdo!

\boldsymbol thank \boldsymbol you \boldsymbol very \boldsymbol much, \boldsymbol dear \boldsymbol friend \boldsymbol mathmax \boldsymbol by \boldsymbol abdo!

Commented by mathmax by abdo last updated on 12/Apr/21

you are welcome sir

you are welcome sir