Question

LaTeX workshop - do not reply

Answer 1

\[\int\limits_{c}^{d} f(x, y,) dy = A(x)\]

Answer 2

\[\pi r^2\]

Answer 3

\[\int\limits_{-1}^{1} \pi r^2 dx = \int\limits_{-1}^{1}(\frac{ x+1 }{ 2 })^2 dx = \frac{ 1 }{ 2 } \int\limits_{-1}^{1} (x^2 + 2x + 1)dx\]

Answer 4

\[\int\limits\limits_{-1}^{1} \pi r^2 dx = \int\limits\limits_{-1}^{1}\pi(\frac{ x+1 }{ 2 })^2 dx = \frac{ \pi }{ 4 } \int\limits\limits_{-1}^{1} (x^2 + 2x + 1)dx\]

Answer 5

\[ \frac{ \pi }{ 4 } \int\limits\limits_{-1}^{1} (x^2 + 2x + 1)dx = \frac{\pi}{4}\int\limits\limits_{-1}^{1} x^2 dx + \frac{\pi}{4}\int\limits\limits_{-1}^{1} 2x dx + \frac{\pi}{4}\int\limits\limits_{-1}^{1} 1dx\]

Answer 6

\[=\frac{\pi}{12}x^3|_{-1}^1+\frac{\pi}{4}x^2|_{-1}^1+\frac{\pi}{4}x|_{-1}^1=\frac{ \pi }{ 6 }+0+\frac{ \pi }{ 2 }=\frac{4\pi}{6}=\frac{2\pi}{3}\]