Question

Помогите по алгебре 100 балов ​

Answer 1

Объяснение:

1.

$y=2x-x^2\ \ \ \ y=0\ \ \ \ OX \ \ \ \ V=?\\2x-x^2=0\\x^2-2x=0\\x*(x-2)=0\\x_1=0\ \ \ \ \\x-2=0\\x_2=2.\\V=\pi *\int\limits^2_0 {(2x-x^2-0)^2} \, dx =\pi *\int\limits^2_0 {(2x-x^2)^2} \, dx =\pi *\int\limits^2_0 {(4x^2-4x^3+x^4)} \, dx =\\=\pi *(\int\limits^2_0 {4x^2 } \, dx -\int\limits^2_0 {4x^3} \, dx +\int\limits^2_0 {x^4} \, dx )=\pi *(\frac{4x^3}{3} \ |_0^2-\frac{4x^4}{4} \ |_0^2+\frac{x^5}{5}\ |_0^2) =$

$=\pi *(\frac{4*2^3}{3}-\frac{4*0^3}{3} -(2^4-0^4)+\frac{2^5}{5}-\frac{0^5}{5})=\pi *(\frac{32}{3} -16+\frac{32}{5} )=\pi *(\frac{160+96}{15}-16)=\\=\pi* (\frac{256}{15} -16)= \pi *(17\frac{1}{15}-16)=\pi *1\frac{1}{15}=\frac{16\pi }{15}.$

Ответ: V=3,35103 куб. ед.

2.

$2x-y-2=0\ \ \ \ y=0\ \ \ \ x=3\ \ \ \ OX\ \ \ \ V=?\\y=2x-2\\2x-2=0\ |:2\\x-1=0\\x=1.\\V=\pi *\int\limits^3_1 {(2x-2)^2} \, dx =\pi *\int\limits^3_1 {(2*(x-1))^2} \, dx =4\pi *\int\limits^3_1 {(x-1)^2} \, dx=\\=4\pi *\int\limits^3_1 {(x^2-2x+1)} \, dx =4\pi *(\int\limits^3_1 {x^2} \, dx -\int\limits^3_1 {2x} \, dx+\int\limits^3_1 {1} \, dx )=\\=4\pi *(\frac{x^3}{3} \ |_1^3-x^2\ |_1^3+x\ |_1^3)=4\pi *(\frac{3^3}{3}-\frac{1^3}{3} -(3^2-1^2)+3-1 )=$

$=4\pi *(9-\frac{1}{3} -8+2)=4\pi *2\frac{2}{3} =4\pi *\frac{8}{3}=\frac{32\pi }{3}.$

Ответ: V=33,5103 куб. ед.