Understanding Engineering Mathematics

10.1.6 Volume of a solid of revolution ➤308 311➤➤

Find the volume of the solid of revolution formed when the positive area enclosed under
the following curve is rotated once about thex-axis.

y= 1 −x^2 ,y= 0 , − 1 ≤x≤ 1

10.2 Revision

10.2.1 The derivative as a gradient and rate of change

➤

291 309➤

The derivative,

dy

dx

, of a functiony=f(x)is defined by a limiting process precisely

to give thegradientorslopeof the curve described by the function at a given point
x(230
➤
). As such it describes therate of changeof the function – the ‘steepness’ of
the curve. Thus, at a point wheredy/dxis large and positive the curve of the func-
tiony=f(x)is increasing steeply asxincreases. Other cases are tabulated below, as
examples – fill in the missing entries.

dy/dx behaviour ofy=f(x)

? yincreases slowly asxincreases

large and negative?

? ydecreases slowly asxincreases

It is important to note that the derivative is definedat a point. Differentiation is therefore
what is called alocaloperation. The derivative of a function will only tell us about the
behaviour at a single point, and says nothing about the overall or global behaviour of the
function.

Solution to review question 10.1.1

(i) For the functiony=x^2 we have

dy

dx

= 2 x

Atx=1 this has the value 2 and so (a) the gradient at this point is

2, and (b) the rate of change is also 2.

(ii) For the functiony=cosxthe derivative is

dy

dx

=−sinxwhich is

equal to zero atx=π. So in this case the (a) gradient or slope of the

graph is zero atx=π, as is (b) the rate of change of the function. So,

at this point the tangent to the curve is horizontal, its slope being zero,

and the rate of change of the function is zero. We know, of course,

from the properties of cosx, that this point is a local minimum (see

Section 10.2.3) (182

➤

).