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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

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)
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
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
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
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%
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
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
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
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 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 =
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
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
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
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
(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
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
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
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
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.
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
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
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
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
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)
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
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
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
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
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
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
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
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
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

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