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Sagot :
Y = 4x*e^(-x) , y=0, x=3 , rotation about y-axis
Graph or sketch the function.
The volumeelement is a cylinder with radius x height y and thickness dx.
Integrate from x=0 to x=3
V = 2 π ∫x*y dx =
2 π ∫x*4x*e^(-x) dx
8 π ∫x²*e^(-x) dx ,
Integrate by part: ∫e^(-x)*x² dx , let u’ = e^(-x) => u = -e^(-x) and v = x² => v’ = 2x
∫e^(-x)*x² dx = -e^(-x)*x² + ∫e^(-x)*2x dx
Integrate by part: ∫e^(-x)*2x dx = -e^(-x)*2x + ∫e^(-x)*2 dx
∫2e^(-x) dx = -2e^(-x).
If we wrap up the above we get :
V = 8 π*[-e^(-x)*(x²+2x+2)]
V = 8 π*[e⁻³(9+6+2) – (-e⁰ *2)]
V = 8 π*[17*e⁻³+ 2] = 28.99363 v.u.
Graph or sketch the function.
The volumeelement is a cylinder with radius x height y and thickness dx.
Integrate from x=0 to x=3
V = 2 π ∫x*y dx =
2 π ∫x*4x*e^(-x) dx
8 π ∫x²*e^(-x) dx ,
Integrate by part: ∫e^(-x)*x² dx , let u’ = e^(-x) => u = -e^(-x) and v = x² => v’ = 2x
∫e^(-x)*x² dx = -e^(-x)*x² + ∫e^(-x)*2x dx
Integrate by part: ∫e^(-x)*2x dx = -e^(-x)*2x + ∫e^(-x)*2 dx
∫2e^(-x) dx = -2e^(-x).
If we wrap up the above we get :
V = 8 π*[-e^(-x)*(x²+2x+2)]
V = 8 π*[e⁻³(9+6+2) – (-e⁰ *2)]
V = 8 π*[17*e⁻³+ 2] = 28.99363 v.u.
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