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Chapter 14: Q 19. (page 1106)

In Chapter 4 we defined the average value of an integrable function fon the interval [a,b]to be 1b-a鈭�abf(x)dx.

Generalize this expression to a formula that will give the average value of f(x,y,z) along a smooth curve Cparametrized by r(t) fora鈮�t鈮�b.

Short Answer

Expert verified

The parametrized equation isfavg(C)=鈭�Cf(x,y,z)ds鈭�Cds.

Step by step solution

01

Given Information

It is given that favg=1b-a鈭�abf(x)dx

We need to generalize the expression to find average value of fx,y,zalong a smooth curve parametrized byrtfora鈮�t鈮�b.

02

Simplifying average value of function

Since favg=1b-a鈭�abf(x)dx

=鈭�abf(x)dxb-a

=鈭�abf(x)dx鈭�abdx

03

Deducing the formula for average value

The value of integral is along smooth curve is

鈭�Cf(x,y,z)ds

The average value is

role="math" localid="1653588021219" favg(C)=鈭�Cf(x,y,z)dsArc Length ofCfora鈮�t鈮�b

=鈭�Cf(x,y,z)ds鈭�Cds

Hence, the formula for average value of fx,y,zalong smooth curve parametrized by r(t)for a鈮�t鈮�bis

favg(C)=鈭�Cf(x,y,z)ds鈭�Cds

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