Question

Sa se calculeze volumul corpurilor de rotație determinate de functiile f (x)=4-x, aparținând [0,4]

Answer 1

$\pi \int\limits^4_0 {f^2(x)} \, dx =$

$f^2(x)=(4-x)^2=16-8x+x^2$

[tex] \pi [\int\limits^4_0 16 -8x+x^2 ] dx=[/tex]

[tex]\pi [\int\limits^4_0 16 -8x+x^2 ]= \pi [16 \int\limits^4_0 dx -8 \int\limits^4_0 x dx+ \int\limits^4_0 x^2 dx]= \pi [16x-8 \frac{x^2}{2}+ \frac{x^3}{3}] [/tex]

[tex] \pi [16x-8 \frac{x^2}{2}+ \frac{x^3}{3}] = \pi [16x-4 x^2+ \frac{x^3}{3}] = [/tex]

$\pi [64-64+ \frac{64}{2} ] = \pi [\frac{64}{2}] = 32 \pi$