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modern engineering mathematics

Modern Engineering Mathematics 6th Edition Glyn James - Solutions

Using the definition of an integral (8.35), show that S's a (x - 1)dx 1)dx = (ba) (b - a) -
What is the volume of a pyramid with square base, of side 4 metres and height 6 metres?
A reservoir is created by constructing a dam across a glacial valley. Its wet face is vertical and has approximately the shape of a parabola, as shown in Figure 8.50. The water pressure p (pascals) varies with depth according towhere p0 is the pressure at the surface, g is the acceleration due to
A beam of length l is freely hinged at both ends and carries a distributed load w(x) whereShow that the total load is W and find the shear force at a point on the beam. w(x) = [4Wx/12 0x1/2 4W(1 x)/1 1/2 x 1
Find the indefinite integrals of (a) 6x4 + 4x 3 - X (b) (2-x)x (c) (5x + 2) (d) x + 1 X
Evaluate the definite integrals (a) (c) 2 for (x4 6x 4)dx (b) -2 4edx - 2 (d) 1 (x - 1) x TC/6 [ 0 dx (cos 3x + 2 sin 3x)dx
Using the inverse-function rule, obtain the integrals of(a) sin–1x (b) ln x
Evaluate the following integrals. (a) 10 [*; (6 - Aydh (6) [" 0 0 310(po+ 10ggy)(y)dy
(a) An object moves along a straight line. Its displacement from its initial position is s(t). Show that its velocity v(t) is given by s´(t). The acceleration of the object is a(t). Show thatand deduce that a(t) = s´´(t).(b) A ball-bearing travels along a track with velocity v(t) m s–1 given
Find the definite integrals (a) So 0 (c) dx (3 - x) 2 S= 0 dx 4+x (b) (d) 2 S 0 5 dx (3 + 2x - x) dx x + 10x + 50
Evaluatewhere H is the Heaviside step function given by (2.45).2.45 (a) -1 |x|dx 10 (b) S 0 H(x - 5)dx
As shown in Example 8.7, the bending moment M and shear force F acting in a beam satisfy the differential equationIn Example 8.42, we showed that for a continuously non-uniformly loaded beam which is freely hinged at both ends the shear force F is given byGiven that M = 0 at x = 0, find an
Find the indefinite integrals of(a) x ln x (b) x2cos x (c) ex sin 2x
Find the indefinite integrals (a) 2x(x + 3)dx a) 2x(x. (b) S x + 1 x + 2x + 2 dx
Using partial fractions, evaluate the integrals (a) 6 x 2x 8 -dx (b) 2 (x 1)(x + 2)? 9 -dx (c) 6 0 1 x2 + 5x + 6 -dx
Find the indefinite integrals of (a) 1 x - 10x + 50 (b) 1 (x + 1)(x + 2x + 2) (c) 3x (x - 1)(x + 2)
Evaluate 2 (1 + x)(1 + x) -dx.
Find the indefinite integrals of(a) cos2x (b) sin(5x + 1)cos(x + 2)
Find the indefinite integrals of(a) sin3x cos2x (b) tan x
Find the indefinite integrals of(a) sinh 5x cosh 2x (b) sech x
Find the indefinite integral 1 2 + (1-x) -dx
Find the indefinite integral ∫√(1 – x2)dx, 0 ≤ x ≤ 1.
Consider the storage tank described in Question 4 of Exercises 2.2.2. Show that the volume V(h) of oil when the depth is h is given byWhat is the value of h when the volume of oil is reduced to 10% of its capacity?Data from Question 4 of Exercises 2.2.2.An oil storage tank has the form of a
Using the substitution u = √(x + 2), evaluate the definite integral 2 -2 (x + 2) x+6 -dx
Find the indefinite integrals of the following functions (a) (x + 6x - 7) (d) 2x + 3 (x + 4x +9) 1 V(r2 5x + 4) (e) x(x + 4x - 3) (b) 1 (3x - 6x + 7) (f) (3 + 2x2x) (c)
Find the length of the perimeter of the ellipse x = a sin t, y = b cos t, 0 ≤ t ≤ 2π.
The area enclosed between the curve y = √(x – 2) and the ordinates x = 2 and x = 5 is rotated through 2 radians about the x axis. Calculate(a) The rotating area and the coordinates of its centroid;(b) The volume of the solid of revolution generated and the coordinates of its centre of gravity.
Show that the volume of a cap of height h of a sphere of radius r is π(3r – h)h2/3.
An electric current i is given by the expression i = I sin θ where I is a constant. Find the root mean square value of the current over the interval 0 ≤ θ ≤ 2π.
A parabolic reflector is formed by rotating the part of the curve y = √x between x = 0 and x = 1 about the x axis. What is the surface area of the reflector?
The curve described by the cable of the suspension bridge shown in Figure 8.70 is given bywhere x is the distance measured from one end of the bridge. What is the length of the cable (see Example 8.5)?Data from Example 8.5In a suspension bridge a roadway, of length 2l, is suspended by vertical
Find the equation of the curve described by a heavy cable hanging, without load, under gravity, from two equally high points.
Find the moments of inertia of a circular disc of radius a about(a) A diameter;(b) An axis through its centre and perpendicular to it.Assume uniform mass per unit area is ρ.
Evaluate the integralto 5dp, using the trapezium rule. f(1/x) dx
Evaluate the integralto 5dp, using the trapezium rule and extrapolation. S(1 + x) dx
Figure 8.77 shows a longitudinal section PQ of rough ground through which a straight horizontal road is to be cut. The width of the road is to be 10 m, and the sides of the cutting and embankment slope at 2 horizontal to 1 vertical. Estimate the net volume of earth removed in making the road.
Find the slope of the tangent to the lemniscateat the point (x, y). (x + y) = a(x - y)
A cylinder of length l and diameter D is constructed such that the density of the material comprising it varies as the distance from the base. Show that the mass of the cylinder is given bywhere K is a proportionality constant. S 0 KDxdx
A beam of length l is freely hinged at both ends and carries a distributed load w(x) whereFind the shear force at a point on the beam. W = [4W/1 0x 1/4 0 1/4
A hemispherical vessel has internal radius 0.5 m. It is initially empty. Water flows in at a constant rate of 1 litre per second. Find an expression for the depth of the water after t seconds.
Using the Fundamental Theorem of Integral and Differential Calculus, evaluate the following integrals: (a) xdx, (b) fedx, (c) sin 5x dx, sin noting thatx7x6 dx (e) sec3x dx, noting that dex dx x = 3ex d noting that cos 5x d.x = -5 sin 5x d (d) - (2x + 1)'dx, noting that (2x + 1) dx 8(2x + 1) d
Find the indefinite integrals of (a) 3x2/3 (c) 2x - 2x + 2 +11/1-2 X (e) x + 3e. 1 x (g) (1-2x)/3 (i) cos(2x + 1) (b) (2x) (d) 2e +3 cos 2x (f) (2x + 1) (h) (2x + 1) (j) 2
Evaluate the definite integrals(Replace the x in (a) by (x + 1) – 1 and in (b) by (x –1) + 1.) S. 2 (a) x dx (x+1) (c) ) f (x - 1) dx 1) dx (e) dx (3 + 2x - x) 0 (b) x(x - 1)dx (d) f* 0 sin.x dx
Find the indefinite integrals of (a) x (c) 4x - 7x +1 x (g) 1 9 - 16x (e)- 1 (1-9x) (b) (x + 1)-1/3 (d) sin x + cos x (f) (h) 1 (2x - x) 1 (4- x)
Evaluate (a) (c) (e) 3 [lx 3 S. 3 |x2|dx [x]dx x[x]dx 0 (b) fax. 0 (d) (x - 2)H(x - 2)d.x 3 S. 0 FRACPT(x)d.x
The function f(x) is periodic with period 1 and is defined on [0, 1] bySketch its graph and obtain the graph offor –4 ≤ x ≤ 4. Show that g(x) is a periodic function of period 1. f(x) = 1 0
Draw the graph of the function f(x) defined byfor –2π f(x): = 0 sin (sin t)dt
Use integration by parts to find the indefinite integrals of(a) x sin x (b) xe3x (c) x3 ln x(d) e–2xsin 3x (e) x tan–1x (f) x cos 2x
Using integration by parts, evaluate the definite integrals (a) xsin xdx (b) 7/2 (c) or 3 1 Jo \ x uzx xpxax 0 $(a) xsin xdx (b) 7/2 (c) or 3 1 Jo \ x uzx xpxax 0$
Use the composite function rule to integrate the following functions: (a) x(1+x) (b) cos x sinx (d) X (x - 1) X (1 + x) (e) (h) 2x + 3 x + 3x + 2 X (4-x) Va (c) x (1 + x) (f) sinx cosx
Find the values of the constants a and b such thatand hence find its integral. (Note that (d/dx)(x2 + 2x + 5) = 2x + 2.) 3x + 2 2 x + 2x + 5 a(2x + 2) 2 x + 2x + 5 + b 2 x + 2x + 5
Use the technique of Question 113 to integrateData from Question 113Find the values of the constants a and b such thatand hence find its integral. (Note that (d/dx)(x2 + 2x + 5) = 2x + 2.) (a) x + 1 x + 4x + 5
Evaluate the following definite integrals with the given substitution: (a) (b) 1/2 1/6 d.x (5 + 6x) tan x with u = 5 + 6x H 1 + x dx, with u = tan^'x 0
Show thatUse this result to integrate(a) sin–1x (b) ln x (c) cosh–1x(d) tan–1x froodx -xf f(x)dx= xf(x) - xf'(x)dx
Using partial fractions, integrate (a) (c) (e) 50 X x-3x - 4 1 x(x +1) 1 x - 1 1 x(x - 1)(x - 2) (b) (d) (h) X (x - 2) 2 X x + 2x + 1 1 x(x - 1) 1 1 + x - 2x
Express 12/(x – 3)(x +1) in partial fractions and hence show that 6 12 (x 3)(x + 1) - - dx = 3 In /5 15
Find the indefinite integrals(a) sin 3x cos 5x (b) cos 7x cos 5x(c) sin2x (d) cos2x(e) cosh2x (f) sinh(5x + 1)
Evaluate the definite integrals T a) ["sin's sin 5x sin 6x dx (b) sin5x dx
Use an appropriate substitution to integrate the following functions: (a) 1 1 + (1 + x) (b) sinx cosx (c) sin Vx
Show that t = tan 1/2x impliesHence integrate and sin x = COS X = d.x= 21 + 1 1-t 1 + 1 1 2 1+t -dt
Evaluate the following definite integral with the given substitution: S -2 x+6 (x + 2) -d.x, with u = (x + 2)
In Question 11 (Exercises 8.2.8) the equation of the path of P was found to be such thatUse the substitution x = a sech u to integrate this differential equation and show thatThis curve is called a tractrix.Data from Question 11A small weight is dragged across a horizontal plane by a string PQ of
Find the indefinite integrals (a) (b) (c) (d) f(3 +2 (e) + 2x - x)dx d.x (x - 6x+5) f - dx dx (x - 4x+8) 9) fx(3 + dx x + 3 (x + 4x +13) -d.x + 2x - x)dx
Find the volume generated when the plane figure bounded by the curve xy = x3 + 3, the x axis and the ordinates at x = 1 and x = 2 is rotated about the x axis through one complete revolution.
Express the length of the arc of the curve y = sin x from x = 0 to x = π as an integral. Also find the volume of the solid generated by revolving the region bounded by the x axis and this arc about the x axis through 2π radians.
(a) Sketch the curve whose equation is y = (x – 2)(x – 1) Show that the volume generated when the finite area between the curve and the x axis is rotated through 2π radians about the x axis is π/30.(b) Show that the curved surface generated by the revolution about the x axis of the portion
A curve is represented parametrically byFind the volume and surface area of the solid of revolution generated when the curve is rotated about the x axis through 2π radians. x(t) = 3tt, y(t) = 3t (0 t1)
The electrical resistance R (in Ω) of a rheostat at a temperature θ (in °C) is given by R = 38(1 + 0.004θ). Find the average resistance of the rheostat as the temperature varies uniformly from 10°C to 40°C.
The area enclosed between the x axis, the curve y = x(2 – x) and the ordinates x = 1 and x = 2 is rotated through 2π radians about the x axis. Calculate(a) The rotating area and the coordinates of its centroid;(b) The volume of the solid of revolution formed and the coordinates of its centre of
Show that the area enclosed between the x axis, the curve 4y = x2 – 2 ln x and the coordinates x = 1 and x = 3 is 1/6(19 – 9 ln 3).
The speed V of a rocket at a time t after launch is given by V = at2 + b where a and b are constants. The average speed over the first second was 10ms–1, and that over the next second was 50ms–1. Determine the values of a and b. What was the average speed over the third second?
Find the centroid of the area bounded by y2 = 4x and y = 2x and also the centroid of the volume obtained by revolving this area about the x axis.
Show that the moment of inertia of an equilateral triangular lamina of side 2a about an altitude is ma2/6, where m is the mass of the lamina.
Use the trapezium rule to evaluateTake the step size h equal to 0.8, 0.4, 0.2, 0.1 in turn and use extrapolation to improve the accuracy of your answer. 0.8edx. Sex
Use the trapezium rule, with interval-halving and extrapolation, to evaluate 0 log(cosh x) dx to 4dp
An ellipse has parametric equations x = cos t, y = 1/2 √3 sin t. Show that the length of its circumference is given byThis integral cannot be evaluated in terms of elementary functions. Use the trapezium rule with interval-halving to evaluate it to 6dp. p/2 25" (3 + sint)dt 0
The capacity of a battery is measured by ∫i dt, where i is the current. Estimate, using Simpson’s rule, the capacity of a battery whose current was measured over an 8 h period with the results shown below: 8 Time/h 0 1 2 3 4 5 6 7 Current/A 25.2 29.0 31.8 36.5 33.7 31.2 29.6 27.3 28.6
The speed V(t)ms–1 of a vehicle at time t s is given by the table below. Use Simpson’s rule to estimate the distance travelled over the 8 seconds. 0 1 2 3 4 5 6 7 8 V(t) 0 0.63 2.52 5.41 9.02 13.11 16.72 18.75 20.15
Use Simpson’s rule with h = 0.1 to estimate S. 0 (1 + x)dx
An isosceles trapezium has the dimensions, expressed in cms, shown in Figure 8.1. Find the formula which relates the enclosed area A cm2 to the height x cm. Evaluate A for x = 1, 2, 3, 4 and 5. 2x X Figure 8.1 Trapezium 10 X 2x
Sketch the graph of the cubic function with formula y = x3 + 3x2 − 3x + 4
Find the general solution of the linear recurrence relation (n+ 1)xn+1 = nx = 1, for n = 1
Evaluate the expression 2xn + 2 – 7xn + 1 + 3xn when xn is defined for all n ≥ 0 byWhich of (a) to (d) are solutions of the following recurrence relation? (a) x = 3" (c) x=2-" Xn- (b) x=2" (d) x,,= 3(-2)" Xn
Show that the seriesis divergent. + + -|2
Find the radius of convergences of the series (2) n=| n (b) n"x" n=l
Using the definition of a derivative given in (8.1), find f'(x) when f(x) is(a) A constant K (b) x (c) x2 – 2(d) x3 (e) √x (f) 1/(1 + x)
Consider the function f(x) = 2x2 – 5x – 12. Find(a) The derivative of f(x) from first principles;(b) The rate of change of f(x) at x = 1;(c) The points at which the line through (1, –15) with slope m cuts the graph of f(x);(d) The value of m such that the points of intersection found in (c)
Consider the function f(x) = 2x3 – 3x2 + x + 3. Find(a) The derivative of f(x) from first principles;(b) The rate of change of f(x) at x = 1;(c) The points at which the line through (1, 3) with slope m cuts the graph of f(x);(d) The values of m such that two of the points of intersection found in
Show from first principles that the derivative ofHence confirm the result outlined in blue in Section 2.3.4 and using the calculus method verify the results of this example.Data from Section 2.3.4 is f(x) = ax+bx+ c f'(x) = 2ax + b
Show that if f(x) = ax3 + bx2 + cx + d, thenDeduce that f’(x) = 3ax2 + 2bx + c f(x+Ax) = ax + bx + cx+d+(3ax +2bx+c)Ax + (3ax + b)(Ax) + a(Ax)
The displacement–time graph for a vehicle is given byObtain the formula for the velocity–time graph. s(t)= t-t+1, 3t - 3, 19-t, 0t1 1
Consider the function f(x) = √(1 + sin x). Show that f(3π/2 ± h) = √2 sin 1/2 h (h > 0) and deduce that f´(x) does not exist at x = 3π/2.
Gas escapes from a spherical balloon at 2m3min–1. How fast is the surface area shrinking when the radius equals 12 m? (The surface area of a sphere of radius r is 4πr2.)
A tank is initially filled with 1000 litres of brine, containing 0.15 kg of salt per litre. Fresh brine containing 0.25 kg of salt per litre runs into the tank at a rate of 4 litres s–1, and the mixture (kept uniform by vigorous stirring) runs out at the same rate. Show that if Q (in kg) is the
The bending moment M(x) for a beam of length l is given by M(x) = W(2x – l)3/8l2, 0 ≤ x ≤ l. Find the formula for the shear force F.
A small weight is dragged across a horizontal plane by a string PQ of length a, the end P being attached to the weight while the end Q is made to move steadily along a fixed line perpendicular to the original position of PQ. Choosing the coordinate axes so that Oy is that fixed line and Ox passes
The limiting tension in a rope wound round a capstan (that is, the tension when the rope is about to slip) depends on the angle of wrap θ, as shown in Figure 8.15. Show that an increase Δθ in the angle of wrap produces a corresponding increase ΔT in the value of the limiting tension such
A chemical dissolves in water at a rate jointly proportional to the amount undissolved and to the difference between the concentration in the solution and that in the saturated solution. Initially none of the chemical is dissolved in the water. Show that the amount x(t) of undissolved chemical
The rate at which a solute diffuses through a membrane is proportional to the area and to the concentration difference across the membrane. A solution of concentration C flows down a tube with constant velocity v. The solute diffuses through the wall of the tube into an ambient solution of the same
A lecture theatre having volume 1000m3 is designed to seat 200 people. The air is conditioned continuously by an inflow of fresh air at a constant rate V (in m3min–1). An average person generates 980cm3 of CO2 per minute, while fresh air contains 0.04% of CO2 by volume. Show that the percentage

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