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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Use the most convenient method (Disk or Shell Method) to find the volume obtained by rotating region B in Figure 12 about the given axis.x = 2 6 2 y A y = x² + 2 B 2 X
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = sin x, y = csc2x, x = π/4
Let R be the intersection of the circles of radius 1 centered at (0, 0) and (0, 1). Express an integral that gives the volume of revolution of R about the x-axis. (Do not evaluate the integral.)
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis. 2 NIE x = sin y, x==y π
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.y = 2 √x, y = x, about y = 4
Use the most convenient method (Disk or Shell Method) to find the volume obtained by rotating region B in Figure 12 about the given axis.x-axis 6 2 y A y=x² +2 B 2 X
Calculate the average over the given interval. f(x) = X (x² + 16)3/2¹ [0, 3]
Calculate the average over the given interval. f(x) = n sinnx, [0,
Let a > 0. Show that the volume obtained when the region between y = a √x − ax2 and the x-axis is rotated about the x-axis is independent of the constant a.
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.y = x3, y = x1/3, for x ≥ 0, about y-axis
Use the most convenient method (Disk or Shell Method) to find the volume obtained by rotating region B in Figure 12 about the given axis.y = −2 6 2 y A y=x² +2 B 2 X
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = sin x, y = x sin(x2), 0 ≤ x ≤ 1
Calculate the average over the given interval.ƒ(x) = sin(nx), [0, π]
If 12 J of work are needed to stretch a spring 20 cm beyond equilibrium, how much work is required to compress it 6 cm beyond equilibrium?
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis. y = sin(√x) y = 0, ² ≤x≤ 9n²
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.y = x2, y = x1/2, about x = −2
The temperature (in degrees Celsius) at time t (in hours) in an art museum varies according to T(t) = 20 + 5 cos (π1/2 t). Find the average over the time periods [0, 24] and [2, 6].
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. y = 9 y = 10 x², x ≥ 0, about y = 12 -
A spring whose equilibrium length is 15 cm exerts a force of 50 N when it is stretched to 20 cm. Find the work required to stretch the spring from 22 to 24 cm.
Plot on the same set of axes. Use a computer algebra system to find the points of intersection numerically and compute the area between the curves. y= X Vx2 + 1 and_y = ( x 1 ) 2 -
Use the most convenient method (Disk or Shell Method) to find the given volume of rotation.Region between x = y(5 − y) and x = 0, rotated about the y-axis
Use the most convenient method (Disk or Shell Method) to find the given volume of rotation.Region between x = y(5 − y) and x = 0, rotated about the x-axis
A steel bar of length 3 m experiences extreme heat at its center, so that the temperature at coordinate x on the bar is given by T(x) = 40 sin (πx/3) + 50°C where the bar sits along the interval [0, 3] on the x-axis. Determine the average temperature of the bar.
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. y = 9 x2 y = 10 x², x≥ 0, about x = -1
If 18 ft-lb of work are needed to stretch a spring 1.5 ft beyond equilibrium, how far will the spring stretch if a 12-lb weight is attached to its end?
Sketch a region whose area is represented byand determine the area using geometry. √2/2 -√2/2 xp (|x|-zx-IN)
Use the most convenient method (Disk or Shell Method) to find the given volume of rotation.Region bounded by y = x2 and x = y2, rotated about the y-axis
The temperature in the town of Walla Walla during the month of July follows a pattern given by T(t) = 10 sin(tπ/31) + 14 sin (tπ/2) + 73°F. Here, t is measured in days, and there are 31 days in July. Explain why you might see a pattern like this and compute the average temperature during the
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. y= 1 X y = 5 2 x, about y-axis
Let W be the work (against the sun’s gravitational force) required to transport an 80-kg person from Earth to Mars when the two planets are aligned with the sun at their minimal distance of 55.7 × 106 km. Use Newton’s Universal Law of Gravity to express W as an integral and evaluate it. The
Beginning at the same time and location, Athletes 1 and 2 run for 30 seconds along a straight track with velocities v1(t) and v2(t) (in meters per second) as shown in Figure 23.(a) What is represented by the area of the shaded region over [0, 10]?(b) Which of the following is represented by the
Use the most convenient method (Disk or Shell Method) to find the given volume of rotation.Region bounded by y = x2 and x = y2, rotated about x = 3
The door to the garage is left open and over the next 4 hours, the temperature in a house in degrees Celsius is given by T(t) = 20/(1 + 0.25t)2. Determine the average temperature over those 4 h.
Water is pumped into a spherical tank of radius 2 m from a source located 1 m below a hole at the bottom (Figure 5). The density of water is 1000 kg/m3.Calculate the work required to fill the tank. 2 Water source
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.y2 = 4x, y = x, about y = 8
Express the area (not signed) of the shaded region in Figure 24 as a sum of three integrals involving ƒ(x) and g(x). w. 5 y=f(x), y = g(x)
Use the most convenient method (Disk or Shell Method) to find the given volume of rotation.Region in Figure 13, rotated about the x-axis y ux-x=x 1 x-
Use the most convenient method (Disk or Shell Method) to find the given volume of rotation.Region in Figure 13, rotated about the y-axis y y=x-x¹² 1 X
Find the area enclosed by the curves y = c − x2 and y = x2 − c as a function of c. Find the value of c for which this area is equal to 1.
A ball thrown in the air vertically from ground level with initial velocity 18 m/s has height h(t) = 18t − 9.8t2 at time t (in seconds). Find the average height and the average speed over the time interval extending from the ball’s release to its return to ground level.
The region between the graphs of f and g over [0, 1] is revolved about the line y = −3. Use the midpoint approximation with values from the following table to estimate the volume V of the resulting solid: 0.1 0.3 0.5 7 6 X f(x) 8 g(x) 2 3.5 0.7 0.9 7 8 2 4 3.5
A tank of mass 20 kg containing 100 kg of water (density 1000 kg/m3) is raised vertically at a constant speed of 100 m/minute for 1 min, during which time it leaks water at a rate of 40 kg/min. Calculate the total work performed in raising the container.
Set up (but do not evaluate) an integral that expresses the area between the circles x2 + y2 = 2 and x2 + (y − 1)2 = 1.
Set up (but do not evaluate) an integral that expresses the area between the graphs of y = (1 + x2)−1 and y = x2.
Use the most convenient method (Disk or Shell Method) to find the given volume of rotation.Region in Figure 14, rotated about x = 4 y y=4-x² 1 y=x³ +2 2 X
Find the average speed over the time interval [1, 5] (time in seconds) of a particle whose position at time t is s(t) = t3 − 6t2 m.
You assist your grandfather Umberto who wants to know the volume of his wine barrels. Knowing that you are taking a calculus course, he thought that you might be able to help. So he measured the circumference around each barrel at regular intervals from the bottom to the top and provided the
Find a numerical approximation to the area above y = 1 − (x/π) and below y = sin x (find the points of intersection numerically).
Use the most convenient method (Disk or Shell Method) to find the given volume of rotation.Region in Figure 14, rotated about y = −2 y=4-x² 1 y = x³ +2 2 X
An object with zero initial velocity accelerates at a constant rate of 10 m/s2. Find its average velocity during the first 15 seconds.
You assist your grandfather Umberto who wants to know the volume of his wine barrels. Knowing that you are taking a calculus course, he thought that you might be able to help. So he measured the circumference around each barrel at regular intervals from the bottom to the top and provided the
Find a numerical approximation to the area above y = |x| and below y = cos x.
The acceleration of a particle is a(t) = 60t − 4t3 m/s2. Compute the average acceleration and the average speed over the time interval [2, 6], assuming that the particle’s initial velocity is zero.
Use the Shell Method to find the given volume of rotation.A sphere of radius r
Find the volume of the cone obtained by rotating the region under the segment joining (0, h) and (r, 0) about the y-axis.
Use a computer algebra system to find a numerical approximation to the number c (besides zero) in [0, π/2], where the curves y = sin x and y = tan2 x intersect. Then find the area enclosed by the graphs over [0, c].
What is the average area of circles whose radii vary from 0 to R?
The torus (doughnut-shaped solid) in Figure 15 is obtained by rotating the circle (x − a)2 + y2 = b2 around the y-axis (assume that a > b). Show that it has volume 2π2ab2. After simplifying it, evaluate the integral by interpreting it as the area of a circle. a a+b -X
Use the Shell Method to find the given volume of rotation.The torus obtained by rotating the circle (x − a)2 + y2 = b2 about the y-axis, where a > b (compare with Exercise 60 in Section 6.3). Evaluate the integral by interpreting part of it as the area of a circle.Data From Exercise 60 in
Let M be the average value of ƒ(x) = x4 on [0, 3]. Find a value of c in [0, 3] such that ƒ(c) = M.
Referring to Figure 1 at the beginning of this section, estimate the projected number of additional joules produced in the years 2009–2030 as a result of government stimulus spending in 2009–2010. One watt (W) is equal to 1 joule/second (J/s), and 1 gigawatt (GW) is 109 watts. Gigawatts
Sketch the hypocycloid x2/3 + y2/3 = 1 and find the volume of the solid obtained by revolving it about the x-axis.
Let ƒ(x) = √x. Find a value of c in [4, 9] such that f (c) is equal to the average of ƒ on [4, 9].
The solid generated by rotating the region between the branches of the hyperbola y2 − x2 = 1 about the x-axis is called a hyperboloid (Figure 16). Find the volume of the hyperboloid for −a ≤ x ≤ a. -a 1 -1 y a -x
Use the Shell Method to find the given volume of rotation.The “paraboloid” obtained by rotating the region between y = x2 and y = c (c > 0) about the y-axis
A “bead” is formed by removing a cylinder of radius r from the center of a sphere of radius R (Figure 17). Find the volume of the bead with r = 1 and R = 2. R h
Let M be the average value of ƒ(x) = x3 on [0, A], where A > 0. Which theorem guarantees that ƒ(c) = M has a solution c in [0, A]? Find c.
Find the volume V of the bead (Figure 17) in terms of r and R. Then show that V = π/6 h3, where h is the height of the bead. This formula has a surprising consequence: Since V can be expressed in terms of h alone, it follows that two beads of height 1 cm, one formed from a sphere the size of an
Use the Shell Method to find the given volume of rotation.The “paraboloid” obtained by rotating the region between y = x2 and y = c (c > 0) about the y-axis
Let ƒ(x) = 2 sin x − x. Use a computer algebra system to plot ƒ and estimate:(a) The positive root α of ƒ(b) The average value M of ƒ on [0, α](c) A value c ∈ [0, α] such that ƒ (c) = M
The surface area of a sphere of radius r is 4πr2. Use this to derive the formula for the volume V of a sphere of radius R in a new way.(a) Show that the volume of a thin spherical shell of inner radius r and thickness Δr is approximately 4πr2Δr.(b) Approximate V by decomposing the sphere of
The solid generated by rotating the region inside the ellipse with equation (x/a)2 + (y/b)2 = 1 around the x-axis is called an ellipsoid. Show that the ellipsoid has volume 4/3πab2. What is the volume if the ellipse is rotated around the y-axis?
Find the line y = mx that divides the area under the curve y = x(1 − x) over [0, 1] into two regions of equal area.
The curve y = ƒ(x) in Figure 18, called a tractrix, has the following property: The tangent line at each point (x, y) on the curve has slopeLet R be the shaded region under the graph of y = ƒ(x) for 0 ≤ x ≤ a in Figure 18. Compute the volume V of the solid obtained by revolving R around the
Show that the solid (an ellipsoid) obtained by rotating the region R in Figure 15 about the y-axis has volume 4/3 πa2b. y b R a X
Which of ƒ(x) = x sin2 x and g(x) = x2 sin2 x has a larger average value over [0, 1]? Over [1, 2]?
Let c be the number such that the area under y = sin x over [0, π] is divided in half by the line y = cx (Figure 27). Find an equation for c and solve this equation numerically using a computer algebra system. y 1 y = sinx ·E|N y = cx π X
Let R be the region in the unit circle lying above the cut with the line y = mx + b (Figure 20). Assume that the points where the line intersects the circle lie above the x-axis. Use the method of Exercise 65 to show that the solid obtained by rotating R about the x-axis has volume V = π/6 hd2,
The bell-shaped curve y = ƒ(x) in Figure 16 satisfies dy/dx = −xy. Use the Shell Method and the substitution u = ƒ(x) to show that the solid obtained by rotating the region R about the y-axis has volume V = 2π(1 − c), where c = ƒ(a). Observe that as c → 0, the region R becomes infinite
Find the average of ƒ(x) = ax + b over the interval [−M, M], where a, b, and M are arbitrary constants.
Give an example of a function (necessarily discontinuous) that does not satisfy the conclusion of the MVT for Integrals.
Determine the intervals on which the function is concave up or down and find the points of inflection.y = θ + sin2 θ, [0, π]
Investigate the behavior and sketch the graph of y = 12 − 5x − 2x2.
Find the linearization at the point indicated.V(h) = 4h(2 − h)(4 − 2h), a = 1
Approximate to three decimal places using Newton’s Method and compare with the value from a calculator.√11
Find a point c satisfying the conclusion of the MVT for the given function and interval. Then draw the graph of the function, the secant line between the endpoints of the graph and the tangent line at (c, ƒ (c)), to see that the secant and tangent lines are, in fact, parallel. f(x) = 1 1 + x² [0,
Determine the intervals on which the function is concave up or down and find the points of inflection.y = x(x − 8 √x) (x ≥ 0)
Investigate the behavior and sketch the graph of ƒ(x) = x3 − 3x2 + 2. Include the zeros of ƒ, which are x = 1 and 1 ± √3 (approximately −0.73, 2.73).
Find the linearization at the point indicated.P(θ) = sin(3θ + π), a = π/3
Approximate to three decimal places using Newton’s Method and compare with the value from a calculator.51/3
Find a point c satisfying the conclusion of the MVT for the given function and interval. Then draw the graph of the function, the secant line between the endpoints of the graph and the tangent line at (c, ƒ (c)), to see that the secant and tangent lines are, in fact, parallel.y = √x, [0, 3]
Determine the intervals on which the function is concave up or down and find the points of inflection.y = x7/2 − 35x2
Show that ƒ(x) = x3 − 3x2 + 6x has a point of inflection but no local extreme values. Sketch the graph.
Find the linearization at the point indicated. = tan (x (1-₂)). R(t) a= 4
Approximate to three decimal places using Newton’s Method and compare with the value from a calculator.27/3
Let ƒ(x) = x5 + x2. The secant line between (0, 0) and (1, 2) has slope 2 (check this), so by the MVT, ƒ'(c) = 2 for some c ∈ (0, 1). Plot f and the secant line on the same axes. Then plot y = 2x + b for different values of b until the line becomes tangent to the graph of ƒ. Zoom in on the
Determine the intervals on which the function is concave up or down and find the points of inflection.y = (x − 2)(1 − x3)
Investigate the behavior and sketch the graphs of y = x2/3 and y = x4/3.
Approximate to three decimal places using Newton’s Method and compare with the value from a calculator.3−1/4
Use the Linear Approximation. A bond that pays $10,000 in 6 years is offered for sale at a price P. The percentage yield Y of the bond isVerify that if P = $7500, then Y = 4.91%. Estimate the drop in yield if the price rises to $7700. 100((' Y = 100 10,000) P 1/6 - 1

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