Calculus: Early Transcendentals, Chapter 6, 6.3, Section 6.3, Problem 21

The shell has the radius x, the cricumference is 2pi*x and the height is x*e^(-x) , hence, the volume can be evaluated, using the method of cylindrical shells, such that:
V = 2pi*int_(x_1)^(x_2) x*x*e^(-x) dx
You need to find the next endpoint, using the equation x*e^(-x) = 0 => x = 0
V = 2pi*int_0^2 x^2*e^(-x) dx
You need to use integration by parts to evaluate the volume, such that:
int udv = uv - int vdu
u = x^2 => du = 2xdx
dv = e^(-x) => v = -e^(-x)
int_0^2 x^2*e^(-x) dx = -x^2*e^(-x)|_0^2 + 2int_0^2 x*e^(-x)dx
You need to use integration by parts to evaluate the integral int_0^2 x*e^(-x)dx.
u = x => du = dx
dv = e^(-x) => v = -e^(-x)
int_0^2 x*e^(-x)dx = -x*e^(-x)|_0^2 + int_0^2 e^(-x) dx
int_0^2 x*e^(-x)dx = -x*e^(-x)|_0^2 - e^(-x)|_0^2
int_0^2 x*e^(-x)dx = -2*e^(-2) - e^(-2) +0*e^(0)+ e^(0)
int_0^2 x*e^(-x)dx = -2/(e^2) - 1/(e^2) + 1
int_0^2 x*e^(-x)dx = -3/(e^2)+ 1
int_0^2 x^2*e^(-x) dx = -x^2*e^(-x)|_0^2 + 2(-3/(e^2)+ 1)
int_0^2 x^2*e^(-x) dx = -2^2*e^(-2) - 6/(e^2) + 2
int_0^2 x^2*e^(-x) dx = -4/(e^2) -6/(e^2) + 2
int_0^2 x^2*e^(-x) dx = -10/(e^2) + 2
V = 2pi*(-10/(e^2) + 2)
Hence, evaluating the volume, using the method of cylindrical shells, yields V = 2pi*(-10/(e^2) + 2).

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