Use the shell method to find the volume of the solids generated by revolving the region bounded by the curve and line about the y axis? y=x,y=-x/2,x=2 y=x2,y=2-x,x=0 for x> or equal to 0

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Use the shell method to find the volume of the solids generated by revolving the region bounded by the curve and line about the y axis? y=x,y=-x/2,x=2 y=x2,y=2-x,x=0 for x> or equal to 0

$y = x$ $y=x$
$y = - \frac{x}{2}$ $y=-x/2$
$x = 2$ $x=2$
$y = x^{2}$ $y=x^2$ ,
$y = 2 - x$ $y=2-x$ ,
$x \geq 0$ $x>= 0$

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Answer 1 · 2018-03-29T14:38:22

The shell method uses the formula $2 π \int x \cdot f (x) . d x$ $2pi int x*f(x) color(white)(.) dx$ with $f (x)$ $f(x)$ being our function and $x$ $x$ as our radius

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *$

$y = x$ $y=x$

$2 π \int x \cdot x . d x$ $2pi int x * x color(white)(.)dx$

$2 π \int x^{2} . d x$ $2pi int x^2 color(white)(.) dx$

$2 π \times \frac{x^{2 + 1}}{2 + 1}$ $2pi xx x^(2+1)/(2+1)$ via power rule

$2 π \times \frac{x^{3}}{3}$ $2pi xx x^3/3$

$\frac{2 π x^{3}}{3}$ $(2 pi x^3)/3$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *$

$y = - \frac{x}{2}$ $y = -x/2$

$2 π \int x \cdot - \frac{x}{2} . d x$ $2pi intx * -x/2 color(white)(.)dx$

$\frac{- 2 π}{2} \int x^{2} . d x$ $(-2pi)/2 int x^2color(white)(.) dx$

$- π \int x^{2} . d x$ $-pi intx^2color(white)(.)dx$

$\frac{- π x^{3}}{3}$ $(-pi x^3)/3$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *$

$y = x^{2}$ $y = x^2$

$2 π \int x \cdot x^{2} . d x$ $2pi int x * x^2 color(white)(.) dx$

$2 π \int x^{3} . d x$ $2pi int x^3 color(white)(.) dx$

$2 π \times \frac{x^{4}}{4}$ $2pi xx x^4/4$

$\frac{π x^{4}}{2}$ $(pix^4)/2$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *$

$y = 2 - x$ $y = 2-x$

$2 π \int x (2 - x) . d x$ $2pi int x(2-x) color(white)(.) dx$

$2 π \int 2 x - x^{2} . d x$ $2pi int 2x - x^2 color(white)(.) dx$

$2 π \int 2 x . d x - 2 π \int x^{2} . d x$ $2pi int2x color(white)(.) dx - 2pi int x^2 color(white)(.) dx$

$4 π \int x . d x - \frac{2 π x^{3}}{3}$ $4pi intx color(white)(.) dx - (2pi x^3)/3$

$4 π \frac{x^{2}}{2} - \frac{2 π x^{3}}{3}$ $4pi x^2/2 - (2pix^3)/3$

$2 π x^{2} - \frac{2 π x^{3}}{3}$ $2pi x^2 - (2pix^3)/3$

$\frac{3 (2 π x^{2})}{3} - \frac{2 π x^{3}}{3}$ $(3(2pix^2))/3 - (2pix^3)/3$

$\frac{6 π x^{2} - 2 π x^{3}}{3}$ $(6pix^2 - 2pix^3)/3$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *$

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Use the shell method to find the volume of the solids generated by revolving the region bounded by the curve and line about the y axis? y=x,y=-x/2,x=2 y=x2,y=2-x,x=0 for x> or equal to 0

$y = x$ $y=x$
$y = - \frac{x}{2}$ $y=-x/2$
$x = 2$ $x=2$
$y = x^{2}$ $y=x^2$ ,
$y = 2 - x$ $y=2-x$ ,
$x \geq 0$ $x>= 0$

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Use the shell method to find the volume of the solids generated by revolving the region bounded by the curve and line about the y axis? y=x,y=-x/2,x=2 y=x2,y=2-x,x=0 for x> or equal to 0

y=xy=x y=−x2y=-x/2 x=2x=2 y=x2y=x^2, y=2−xy=2-x, x≥0x>= 0

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$y = x$ $y=x$
$y = - \frac{x}{2}$ $y=-x/2$
$x = 2$ $x=2$
$y = x^{2}$ $y=x^2$ ,
$y = 2 - x$ $y=2-x$ ,
$x \geq 0$ $x>= 0$