ここで勉強すれば数学検定１級の壁は超えられるか。

重積分　予想問題

[math]D=\left\{ \left( x,y\right) \left| \right| y\leqq x\leqq y^{2},1\leqq y\leq 3\right\}[/math]　の条件で

[math]I=\int \int _{D}\sin \left( \dfrac {\pi x}{2y}\right) dxdy[/math]を求めなさい。

[math]I=\int ^{3}_{1}\left( \int ^{y^{2}}_{y}\sin \left( \dfrac {\pi }{2\pi }\right) xdx\right) dy=\int ^{3}_{1}\left[ -\dfrac {2y}{\pi }\cos \left( \dfrac {\pi }{2y}\right) x\right] ^{y^{2}}_{y}dy[/math]

[math]=-\dfrac {2}{\pi }\int ^{3}_{1}y\cos \left( \dfrac {\pi y}{2}\right) dy+0=-\dfrac {2}{\pi }\{ \left[ \dfrac {2}{\pi }y\sin \dfrac {\pi }{2}y\right] ^{3}_{1}-\dfrac {2}{\pi }\int ^{3}_{1}\sin \dfrac {\pi y}{2}dｙ\}[/math]

[math]=-\dfrac {4}{\pi ^{2}}\left( -4-\left[ -\dfrac {2}{\pi }\cos \dfrac {\pi y}{2}\right] ^{3}_{1}\right)= -\dfrac {4}{\pi ^{2}}\left( -4+0\right) = \dfrac {16}{\pi ^{2}}[/math]

答　　[math]\dfrac {16}{\pi ^{2}}[/math]