Calculus: Analytic Geometry and Calculus, with Vectors

262 Integrals

Problems 4.79

1 As suggested by Figure 4.791, let a rod having constant linear density

(mass per unit length) 6 be supposed to be concentrated on the interval a 5 x b

of the x axis. Starting by making a partition of the interval a < x b

calculate M,(1-`b the pth moment about of

a b x

the rod. -Ins.:

Figure 4.791 Af.'!4=p+1i(b-t)P'"1-(a-

2 Using the result of the preceding problem, prove that MX"-'t = 0 if and

only if = lff (a + b).

3 Supposing that

fabf(x)dx=M>0,

show that the constant 9 satisfies the equation

if and only if

fab(x-z)f(x)dx=0

Mx =

fab

xf(x) dx.

Remark: Always remember that, in statistics and elsewhere, x is called the mean

(or mean value) of f over the interval a < x 5 b and that, in mechanics and

elsewhere, x is the x coordinate of a centroid. Remember (or learn) that a

centroid is, as it should be, a point "like a center."

4 Supposing that

fabf(x)dx=M>0

and that the mean (or x coordinate of the centroid) is 9, prove that

M` MM m(x-c)2M.

State the meaning of this formula in words, and use the formula to determine the

value of Z for which M`t has the least possible value. Hint: Start by writing

M() =f ab (x- t)2f(x) dx =f ab [(x- 9) + (x - ))2f(x) dx.

5 Let f be the function for which f(x) = 0 when x < 0 and f(x) = e

when x > 0. Determine and graph the mass function F of which f is the density

function.

6 The density function f defined by the first of the formulas

f()^1

_(X-as)= = 1 e_(t >=

X di

l a v

e 2°' ' F(x) -

2va1_m

has the mass function (or cumulative function) F defined by the second formula.

With the aid of the formula (4.64) make a preliminary attempt to learn the

natures of the graphs of y = f(x) and y = F(x) when M = 0 and v = 0.01.