Zadania z analizy matematycznej.

Oblicz pola ograniczone liniami:

1. y = 5 − x² ,                 y = 2

2. y = x² ,                         y = 2 − x²

3. y = 3 − 2x − x² ,       y = 0

4. y = x² + 4x ,               y = x + 4

5. y = e^x ,                          y = 1 ,                 x = 0

6. y = x² − 4x ,               y = 6 − x² 

7. xy = 6 ,                         x + y = 7 

8. y = x ³ ,                        y = 8 ,                  x = 0 

Oblicz długość łuku krzywej:

1. $\left\{ \begin{matrix} x = t^{2} \\ y = t - \frac{1}{3}t^{3} \\ \end{matrix} \right.\ \ \ \ \ \ \ \ 0 \ll t \ll \sqrt{3\ }$

2. $\left\{ \begin{matrix} x = a\cos^{3}t \\ y = a\sin^{3}\text{t\ } \\ \end{matrix} \right.\ \ \ \ \ \ \ \ \ 0 \ll t \ll \frac{\pi}{2\ }$

3. $\left\{ \begin{matrix} x = a\left( cost + tsin \right) \\ y = a\left( sint + tcos \right) \\ \end{matrix} \right.\ \ $

4. $\left\{ \begin{matrix} x = 2 - 3lnt \\ y = 3 + 2lnt \\ \end{matrix} \right.\ \ \ \ \ \ 1 \ll t \ll e\ $

5. $\left\{ \begin{matrix} x = \left( t^{2} - 2 \right)sint + 2tcost \\ y = \left( 2 - t^{2} \right)cost + 2tsint\ \\ \end{matrix} \right.\ $

6. $\left\{ \begin{matrix} x = \frac{1}{3}t^{3} \\ y = 4 - {\frac{1}{2}t}^{2} \\ \end{matrix} \right.\ $