Understanding Engineering Mathematics

B. Calculate the total signed area between the curves and thex-axis and the limits given
by: (a) geometry, (b) integration.

(i) y= 1 −x, x= 2 , 4 (ii) y=x− 1 ,x=0, 2

C.Evaluate

∫ 1

− 1

|x|dx

D.Find the area enclosed by the curvesy=x^2 +2, andy= 1 −x, and the linesx=0,
x=1.

10.3.6 Volume of a solid of revolution

➤➤

292 308

➤

Determine the volumes obtained by rotating the positive area under each of the following
curves about thex-axis, between the given limits:

(i) y=x( 1 −x),y= 0

(ii) xy=1, y=0, x=2, x= 5

(iii) y=sinx, x=0, x=π

10.4 Applications

1.The major application of differentiation in mechanics is in kinematics. In one-
dimensional motion along thex-direction the displacement of a moving particle is
a function of time,x(t) and at any timet its velocity and acceleration are given
respectively by

v=

dx

dt

= ̇xa=

d^2 x

dt^2

= ̈x

(i) – (iv) give either the positionx(t)(metres) or velocityv(t)(metres per second) of

a particle at timet. In each case find (a) the velocity and (b) acceleration at the point

t=1 sec and (c) determine when the particle is stationary.

(i) x(t)= 3 − 2 t+t^2

(ii) v(t)=

2 t− 1

t^2 + 1

(iii) x(t)=e^2 tsint

(iv) v(t)=cost

2.Sometimes rates of change of a variable are not calculated directly but in terms of the
rate of change of some other variable on which it depends. For example the rate of
change of the area of a circle in terms of the rate of change of its radius. In such cases
we use the function of a function rule:

A=πr^2 so

dA

dt

= 2 πr

dr

dt

and we can calculatedA/dtknowing the rate of change ofr.