Friday, June 22, 2018

Calculus of a Single Variable, Chapter 7, 7.3, Section 7.3, Problem 13

Using the shell method we can find the volume of the solid generated by the given curves,
y = e^(-x^2/2)/sqrt(2pi), y = 0 , x = 0 , x = 1

Using the shell method the volume is given as
V= 2*pi int _a^b p(x) h(x) dx
where p(x) is the function of the average radius =x
and
h(x) is the function of height = e^(-x^2/2)/sqrt(2pi)
and the range of x is given as 0 to 1
So the volume is = 2*pi int _a^b p(x) h(x) dx
= 2*pi int _0^1 (x) (e^(-x^2/2)/sqrt(2pi)) dx
=(2*pi)/(sqrt(2pi)) int _0^1 (x*e^(-x^2/2)) dx
=(2*pi)/(sqrt(2pi)) int _0^1 (x*e^(-x^2/2)) dx
let us first solve
int (x*e^(-x^2/2)) dx
let u = x^2/2
du = 2x/2 dx = xdx
so ,
int (x*e^(-x^2/2)) dx
= int (e^(-u)) du
= -e^(-u) = -e^(-x^2/2)

So, V=(2*pi)/(sqrt(2pi)) int _0^1 (x*e^(-x^2/2)) dx
=(2*pi)/(sqrt(2pi)) [-e^(-x^2/2)]_0^1
=(2*pi)/(sqrt(2pi)) [[-e^(-(1)^2/2)]-[-e^(-0^2/2)]]
=(2*pi)/(sqrt(2pi)) [[-e^(-1/2)]-[-e^(0)]]
=(sqrt(2pi)) [1-[e^(-1/2)]]
= 0.986
is the volume

No comments:

Post a Comment

Why is the fact that the Americans are helping the Russians important?

In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...