Matematika A2a 2008/7. gyakorlat

Ez az szócikk a Matematika A2a 2008 alszócikke.

1.

$f(x,y)=xy\sin(x^2y)\,$

T = [0,1]×[0,π/2]

2.

$f(x,y)=2xy^2e^{x^2y}\,$

T = [0,1]×[0,1]

3.

$f(x,y)=x^7+\dfrac{\mathrm{arctg}(y)}{1+y^2}\,$

T = [0,1]×[0,1]

4.

T = [-1,1] × [0,π/4]

$f(x,y)=\sin(x^3)\frac{1}{\cos^2 y}$

5.

T = [-1,1] × [e,e²]

$f(x,y)=\sin(x^3)\frac{\sin^{2017}(\mathrm{sh}(y))}{\mathrm{ln}\,y}$

6.

T = [a,b] × [c,d]

f(x,y) = g(x)h(y)

téglalapon szeparálható integrandus integrálja szorzattá esik szét:

$\int\limits_{x=0}^b\int\limits_{y=c}^{d}g(x)h(y)\,\mathrm{d}x\,\mathrm{d}y=\int\limits_{x=a}^b g(x)\left(\int\limits_{y=c}^{d} h(y)\,\mathrm{d}y\right)\,\mathrm{d}x=\left(\int\limits_{x=a}^b g(x)\,\mathrm{d}x\right)\cdot\left(\int\limits_{y=c}^{d} h(y)\,\mathrm{d}y\right)$

7.

T = [1,e] × [1,2]

$f(x,y)=\frac{\mathrm{ln}^9\,x}{xy}$

8.

$T=\{(x,y)\mid 0\leq x\leq 1\;\wedge\;0\leq y\leq x^2\}$

$f(x,y)=x^3\cos(xy)\,$

9.

$\int\limits_{y=0}^\pi\int\limits_{0}^{\cos y}x\sin y\;dxdy$

10.

$\int\limits_{\sqrt{\pi}}^{\sqrt{2\pi}}\int\limits_{0}^{x^3}\sin \frac{y}{x}\;dydx$

11.

$\int\limits_{0}^{1}\int\limits_{y}^{1}e^{x^2}\;dxdy$

12.

$\int\limits_{1}^{4}\int\limits_{\sqrt{y}}^{2}\sin\left(\frac{x^3}{3}-x\right)\;dxdy$

13.

$\int\limits_{0}^{2}\int\limits_{1+y^2}^{5}y\cdot e^{(x-1)^2}\;dxdy$

14.

$\iint\limits_{T_{x,y}} xy^7\;dxdy$

$T_{x,y}=\{(x,y)\mid x^2+y^2\leq 1,\quad x\geq 0, \quad y\geq 0\}$

15.

$\iiint\limits_{T_{x,y,z}} \sqrt{x^2+y^2}\cdot xy^2 z^2\;dxdydz$

$T_{x,y,z}=\{(x,y)\mid x^2+y^2\leq 4,\quad x\geq 0, \quad 0\leq z\leq 3\}$

6. gyakorlat

8. gyakorlat