Understanding Engineering Mathematics

(ii) Calculate the moment of inertia about the coordinate axes of

(i) y=x^2 , y=0,x= 4

(ii) 2 y=x^3 ,x= 0 y= 4

(iii) xy=4, x=1,y=0,x= 4

8.The mean value ofy=f(x)betweenx=a,bis given by

1

b−a

∫b

a

f(x)dx.

The RMS value is defined as

√

1

b−a

∫b

a

y^2 dx=

√

1

b−a

∫b

a

(f (x))^2 dx.

It is the square root of the mean value of the square of the function. The RMS value is

very important in alternating current theory – applied to oscillatory currents it has the

effect of averaging over the oscillations (power dissipated is proportional to the RMS

of the current). The RMS is similar to thestandard deviationof statistics.

Find the mean and RMS values of the following functions over the ranges indicated:

(i) y=x(x= 0 , 1 ) (ii) y= 3 x(x= 0 , 1 )

(iii) y=sinx(x= 0 ,π) (iv) y=e−x(x= 0 , 1 )

Answers to reinforcement exercises

10.3.1 The derivative as a gradient and rate of change

A.(i) (a) 2 atx=0, 14 atx=2(b)2atx=0, 14 atx= 2

(ii) (a) 1 atx=0,

1

2

atx=

π

3

(b) 1 atx=0,

1

2

atx=

π

3

(iii) (a) 1 atx=0,e(cos 1−sin 1)atx=1(b)1atx=0,e(cos 1−sin 1)atx= 1

(iv) (a) 0 atx=0,

4

5

atx=2(b)0atx=0,

4

5

atx= 2

B. (−1, 12)

10.3.2 Tangent and normal to a curve

(i) Tangent isy= 4 x−4, normal isx+ 4 y− 1 = 0

(ii) Tangent isy= 4 x−2, normal isx+ 4 y− 9 = 0

(iii) Tangent isy=x−1, normal isx+y− 1 = 0

(iv) Tangent isy=x, normal isy=−x

10.3.3 Stationary points and points of inflection

A.(i) Min at (2,−1)

(ii) Min at (2,−14), max at (−2, 18), point of inflection at (0, 2)