Find the volume of a cone enclosed by the surface formed by rotating about the x-axis the liney = 2x between x = 0 and x = h.
Using (2.46), the volume is given by V =
h
0
π(2x)2dx =
h
0
4πx2dx
=4
3πx3h
0=43π(h3− 0) =43πh3.
As before, it is also possible to form a volume of revolution by rotating a curve about they-axis. In this case the volume enclosed between y = a and y = b is
V =
b a
πx2dy. (2.47)
2.3 EXERCISES
2.10 The functiony(x) is defined by y(x) = (1 + xm)n.
(a) Use the chain rule to show that the first derivative ofy is nmxm−1(1 +xm)n−1. (b) The binomial expansion (see section 1.5) of (1 +z)nis
(1 +z)n= 1 +nz +n(n− 1)
2! z2+· · · +n(n− 1) · · · (n − r + 1) r! zr+· · · . Keeping only the terms of zeroth and first order indx, apply this result twice to derive result (a) from first principles.
(c) Expand y in a series of powers of x before differentiating term by term.
Show that the result is the series obtained by expanding the answer given fordy/dx in (a).
2.11 Show by differentiation and substitution that the differential equation 4x2d2y
dx2− 4xdy
dx+ (4x2+ 3)y = 0 has a solution of the formy(x) = xnsinx, and find the value of n.
2.12 Find the positions and natures of the stationary points of the following functions:
(a) x3− 3x + 3; (b) x3− 3x2+ 3x; (c) x3+ 3x + 3;
(d) sinax with a= 0; (e) x5+x3; (f)x5− x3.
2.13 Show that the lowest value taken by the function 3x4+ 4x3− 12x2+ 6 is−26.
2.14 By finding their stationary points and examining their general forms, determine the range of values that each of the following functionsy(x) can take. In each case make a sketch-graph incorporating the features you have identified.
(a) y(x) = (x− 1)/(x2+ 2x + 6).
(b) y(x) = 1/(4 + 3x− x2).
(c) y(x) = (8 sin x)/(15 + 8 tan2x).
2.15 Show that y(x) = xa2xexpx2 has no stationary points other than x = 0, if exp(−√
2)< a < exp(√ 2).
2.16 The curve 4y3=a2(x + 3y) can be parameterised as x = a cos 3θ, y = a cos θ.
(a) Obtain expressions fordy/dx (i) by implicit differentiation and (ii) in param-eterised form. Verify that they are equivalent.
(b) Show that the only point of inflection occurs at the origin. Is it a stationary point of inflection?
(c) Use the information gained in (a) and (b) to sketch the curve, paying particular attention to its shape near the points (−a, a/2) and (a, −a/2) and to its slope at the ‘end points’ (a, a) and (−a, −a).
2.17 The parametric equations for the motion of a charged particle released from rest in electric and magnetic fields at right angles to each other take the forms
x = a(θ− sin θ), y = a(1− cos θ).
Show that the tangent to the curve has slope cot(θ/2). Use this result at a few calculated values ofx and y to sketch the form of the particle’s trajectory.
2.18 Show that the maximum curvature on the catenaryy(x) = a cosh(x/a) is 1/a. You will need some of the results about hyperbolic functions stated in subsection 3.7.6.
2.19 The curve whose equation isx2/3+y2/3=a2/3for positivex and y and which is completed by its symmetric reflections in both axes is known as an astroid.
Sketch it and show that its radius of curvature in the first quadrant is 3(axy)1/3.
O
C
P Q ρ
ρ
r r + ∆r c
p
p + ∆p
Figure 2.13 The coordinate system described in exercise 2.20.
2.20 A two-dimensional coordinate system useful for orbit problems is the tangential-polar coordinate system (figure 2.13). In this system a curve is defined byr, the distance from a fixed point O to a general point P of the curve, and p, the perpendicular distance fromO to the tangent to the curve at P . By proceeding as indicated below, show that the radius of curvature,ρ, at P can be written in the formρ = r dr/dp.
Consider two neighbouring points,P and Q, on the curve. The normals to the curve through those points meet atC, with (in the limit Q→ P ) CP = CQ = ρ.
Apply the cosine rule to trianglesOP C and OQC to obtain two expressions for c2, one in terms ofr and p and the other in terms of r + ∆r and p + ∆p. By equating them and lettingQ→ P deduce the stated result.
2.21 Use Leibnitz’ theorem to find
(a) the second derivative of cosx sin 2x, (b) the third derivative of sinx ln x,
(c) the fourth derivative of (2x3+ 3x2+x + 2) exp 2x.
2.22 If y = exp(−x2), show thatdy/dx = −2xy and hence, by applying Leibnitz’
theorem, prove that forn≥ 1
y(n+1)+ 2xy(n)+ 2ny(n−1)= 0.
2.23 Use the properties of functions at their turning points to do the following:
(a) By considering its properties near x = 1, show that f(x) = 5x4− 11x3+ 26x2− 44x + 24 takes negative values for some range of x.
(b) Show thatf(x) = tan x− x cannot be negative for 0 ≤ x < π/2, and deduce thatg(x) = x−1sinx decreases monotonically in the same range.
2.24 Determine what can be learned from applying Rolle’s theorem to the following functions f(x): (a) ex; (b) x2+ 6x; (c) 2x2+ 3x + 1; (d) 2x2+ 3x + 2; (e) 2x3− 21x2+ 60x + k. (f) If k =−45 in (e), show that x = 3 is one root of f(x) = 0, find the other roots, and verify that the conclusions from (e) are satisfied.
2.25 By applying Rolle’s theorem toxnsinnx, where n is an arbitrary positive integer, show that tannx + x = 0 has a solution α1 with 0 < α1 < π/n. Apply the theorem a second time to obtain the nonsensical result that there is a realα2in 0< α2< π/n, such that cos2(nα2) =−n. Explain why this incorrect result arises.
2.3 EXERCISES
2.26 Use the mean value theorem to establish bounds in the following cases.
(a) For− ln(1 − y), by considering ln x in the range 0 < 1 − y < x < 1.
(b) Forey− 1, by considering ex− 1 in the range 0 < x < y.
2.27 For the functiony(x) = x2exp(−x) obtain a simple relationship between y and dy/dx and then, by applying Leibnitz’ theorem, prove that
xy(n+1)+ (n + x− 2)y(n)+ny(n−1)= 0.
2.28 Use Rolle’s theorem to deduce that, if the equationf(x) = 0 has a repeated root x1, thenx1is also a root of the equationf(x) = 0.
(a) Apply this result to the ‘standard’ quadratic equationax2+bx + c = 0, to show that a necessary condition for equal roots isb2= 4ac.
(b) Find all the roots off(x) = x3+ 4x2− 3x − 18 = 0, given that one of them is a repeated root.
(c) The equationf(x) = x4+ 4x3+ 7x2+ 6x + 2 = 0 has a repeated integer root.
How many real roots does it have altogether?
2.29 Show that the curvex3+y3− 12x − 8y − 16 = 0 touches the x-axis.
2.30 Find the following indefinite integrals:
(a)
(4 +x2)−1dx; (b)
(8 + 2x− x2)−1/2dx for 2≤ x ≤ 4;
(c)
(1 + sinθ)−1dθ; (d) (x√
1− x)−1dx for 0< x≤ 1.
2.31 Find the indefinite integralsJ of the following ratios of polynomials:
(a) (x + 3)/(x2+x− 2);
(b) (x3+ 5x2+ 8x + 12)/(2x2+ 10x + 12);
(c) (3x2+ 20x + 28)/(x2+ 6x + 9);
(d) x3/(a8+x8).
2.32 Expressx2(ax + b)−1as the sum of powers ofx and another integrable term, and hence evaluate
b/a
0
x2 ax + bdx.
2.33 Find the integralJ of (ax2+bx + c)−1, witha= 0, distinguishing between the cases (i)b2> 4ac, (ii) b2< 4ac and (iii) b2= 4ac.
2.34 Use logarithmic integration to find the indefinite integralsJ of the following:
(a) sin 2x/(1 + 4 sin2x);
(b) ex/(ex− e−x);
(c) (1 +x ln x)/(x ln x);
(d) [x(xn+an)]−1.
2.35 Find the derivative off(x) = (1 + sin x)/ cos x and hence determine the indefinite integralJ of sec x.
2.36 Find the indefinite integrals,J, of the following functions involving sinusoids:
(a) cos5x− cos3x;
(b) (1− cos x)/(1 + cos x);
(c) cosx sin x/(1 + cos x);
(d) sec2x/(1− tan2x).
2.37 By making the substitutionx = a cos2θ + b sin2θ, evaluate the definite integrals J between limits a and b (> a) of the following functions:
(a) [(x− a)(b − x)]−1/2; (b) [(x− a)(b − x)]1/2;
(c) [(x− a)/(b − x)]1/2.
2.38 Determine whether the following integrals exist and, where they do, evaluate them:
(a)
∞
0
exp(−λx) dx; (b)
∞
−∞
x (x2+a2)2dx;
(c)
∞
1
1
x + 1dx; (d)
1
0
1 x2dx;
(e)
π/2
0
cotθ dθ; (f)
1
0
x (1− x2)1/2dx.
2.39 Use integration by parts to evaluate the following:
(a)
y
0
x2sinx dx; (b)
y
1
x ln x dx;
(c)
y
0
sin−1x dx; (d)
y
1
ln(a2+x2)/x2dx.
2.40 Show, using the following methods, that the indefinite integral ofx3/(x + 1)1/2is J =352(5x3− 6x2+ 8x− 16)(x + 1)1/2+c.
(a) Repeated integration by parts.
(b) Settingx + 1 = u2and determiningdJ/du as (dJ/dx)(dx/du).
2.41 The gamma function Γ(n) is defined for all n >−1 by Γ(n + 1) =
∞
0
xne−xdx.
Find a recurrence relation connecting Γ(n + 1) and Γ(n).
(a) Deduce (i) the value of Γ(n + 1) when n is a non-negative integer, and (ii) the value of Γ7
2
, given that Γ1
2
=√ π.
(b) Now, taking factorial m for any m to be defined by m! = Γ(m + 1), evaluate
−32
!.
2.42 DefineJ(m, n), for non-negative integers m and n, by the integral J(m, n) =
π/2
0
cosmθ sinnθ dθ.
(a) EvaluateJ(0, 0), J(0, 1), J(1, 0), J(1, 1), J(m, 1), J(1, n).
(b) Using integration by parts, prove that, form and n both > 1, J(m, n) =m− 1
m + nJ(m− 2, n) and J(m, n) = n− 1
m + nJ(m, n− 2).
(c) Evaluate (i)J(5, 3), (ii) J(6, 5) and (iii) J(4, 8).
2.43 By integrating by parts twice, prove thatInas defined in the first equality below for positive integersn has the value given in the second equality:
In=
π/2
0
sinnθ cos θ dθ =n− sin(nπ/2) n2− 1 . 2.44 Evaluate the following definite integrals:
(a) ∞
0 xe−xdx; (b)1 0
(x3+ 1)/(x4+ 4x + 1) dx;
(c) π/2
0 [a + (a− 1) cos θ]−1dθ with a >12; (d)∞
−∞(x2+ 6x + 18)−1dx.
2.4 HINTS AND ANSWERS
2.45 IfJris the integral
∞
0
xrexp(−x2)dx show that
(a) J2r+1= (r!)/2,
(b) J2r= 2−r(2r− 1)(2r − 3) · · · (5)(3)(1) J0.
2.46 Find positive constantsa, b such that ax≤ sin x ≤ bx for 0 ≤ x ≤ π/2. Use this inequality to find (to two significant figures) upper and lower bounds for the integral
I =
π/2
0
(1 + sinx)1/2dx.
Use the substitutiont = tan(x/2) to evaluate I exactly.
2.47 By noting that for 0≤ η ≤ 1, η1/2≥ η3/4≥ η, prove that 2
3≤ 1 a5/2
a
0
(a2− x2)3/4dx≤π 4.
2.48 Show that the total length of the astroid x2/3+y2/3 = a2/3, which can be parameterised asx = a cos3θ, y = a sin3θ, is 6a.
2.49 By noting that sinhx <12ex< cosh x, and that 1 + z2< (1 + z)2forz > 0, show that, forx > 0, the length L of the curve y = 12ex measured from the origin satisfies the inequalities sinhx < L < x + sinh x.
2.50 The equation of a cardioid in plane polar coordinates is ρ = a(1− sin φ).
Sketch the curve and find (i) its area, (ii) its total length, (iii) the surface area of the solid formed by rotating the cardioid about its axis of symmetry and (iv) the volume of the same solid.
2.4 Hints and answers