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mathematics
calculus

Calculus Early Transcendentals 7th edition James Stewart - Solutions

Use Property 8 of integrals to estimate the value of the integral.
Use the properties of integrals to verify the inequality.a.b.
Use the Midpoint Rule with n = 6 to approximate
Let r(t) be the rate at which the world's oil is consumed, where t is measured in years starting at t = 0 on January 1, 2000, and r(t) is measured in barrels per year. What doesrepresent?
A population of honeybees increased at a rate of r(t) bees per week, where the graph of r is as shown. Use the Midpoint Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks.
LetEvaluate by interpreting the integral as a difference of areas.
Estimate the value of the number c such that the area under the curve y = sinh cx between x = 0 and x = 1 is equal to 1.
If f is a continuous function such thatfor all x, find an explicit formula for f(x).
If f is continuous on [a, b], show that
The following figure shows the graphs of f, f', andIdentify each graph, and explain your choices.
Suppose f is continuous, f(0) = 0, f(1) = 1, f'(x) > 0, and. Find the value of the integral
Evaluate the integral.a.b. c.
IfWhere is a continuous function, find f(4).
a) EvaluateWhere is a positive integer. (b) Evaluate Where and are real numbers with 0 ‰¤ a
Suppose the coefficients of the cubic polynomial P(x) = a + bx + cx2 + dx3 satisfy the Equation a + b/2 + c/3 + d/4 = 0 Show that the equation has a root between 0 and 1. Can you generalize this result for an -degree polynomial?
Prove that if f is continuous, then
Evaluate
If
IfWhere Find f' (Ï€/2)
Evaluate
Find the interval [a, b] for which the value of the integralis a maximum.
State both parts of the Fundamental Theorem of Calculus.
(a) State the Net Change Theorem.(b) If r(t) is the rate at which water flows into a reservoir, what doesrepresent?
Suppose a particle moves back and forth along a straight line with velocity v(t), measured in feet per second, and acceleration .(a) What is the meaning of(b) What is the meaning of(c) What is the meaning of
(a) Explain the meaning of the indefinite integral ˆ« f(x) dx..(b) What is the connection between the definite integraland the indefinite integral ˆ« f(x) dx?
Find the area of the shaded region.a.b.
a. y = 12 - x2, y = x2 - 6 b. y = ex, y = xex, x = 0 c. x = 2y2, x = 4 + y2
Use calculus to find the area of the triangle with the given vertices. (0, 0), (3, 1), (1, 2)
Evaluate the integral and interpret it as the area of a region. Sketch the region.
Use a graph to find approximate -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. a. y = x sin(x2), y = x4 b. y = 3x2 - 2x, y = x3 - 3x + 4
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. a. y = 2/1 + x4, y = x2 b. y = tan2x, = y = √x
Use a computer algebra system to find the exact area enclosed by the curves y = x5 - 6x3 + 4x and y = x.
Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.
A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are, 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wing's cross-section.
Two cars, A and B, start side by side and accelerate from rest.The figure shows the graphs of their velocity functions.(a) Which car is ahead after one minute? Explain.(b) What is the meaning of the area of the shaded region?(c) Which car is ahead after two minutes? Explain.(d) Estimate the time at
The curve with equation y2 = x2(x + 3) is called Tschirnhausen's cubic. If you graph this curve you will see that part of the curve forms a loop. Find the area enclosed by the loop.
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. a. y = ex, y = x2 - 1, x = -1, x = 1 b. y = (x - 2)2, y = x c. y = 1/x, y = 1/x2, x = 2
Find the number such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area.
Find the values of such that the area of the region bounded by the parabolas y = x2 - c2 and y = c2 - x2 is 576.
For what values of m do the line y = mx and the curve y = x / (x2 + 1) y xx 2 1 enclose a region? Find the area of the region.
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. (a) y = 2 - 1/2 x, y = 0, x = 1, x = 2; about the x-axis (b) y = √x - 1, y = 0, x = 5; about the x-axis (c) x = 2√y, x =
Refer to the figure and find the volume generated by rotating the given region about the specified line.(a) (1 about OA(b) (1 about AB(c) (2 about OA
Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. (1) y = e-x2, y = 0, x = -1, x = 1 (a) About the x-axis (b) About y = - 1 (2) x2 +
Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the x-axis the region bounded by these curves. y = 2 + x2 cos x, y = x4 + x + 1
Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = sin2 x, y = 0, 0 < x < π; about y = - 1
Each integral represents the volume of a solid. Describe the solid.(a)(b)
A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square
(a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint Rule with n = 4 to estimate the volume of the solid.(b) Estimate the volume if the region is rotated about the y-axis. Again use the Midpoint Rule with n = 4.
Find the volume of the described solid S.(a) A right circular cone with height h and base radius r(b) A cap of a sphere with radius r and height h(c) A pyramid with height and rectangular base with dimensions b and 2 b
(a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii r and R.(b) By interpreting the integral as an area, find the volume of the torus.
(a) Cavalieri's Principle states that if a family of parallel planes gives equal cross-sectional areas for two solids S1 and S2, then the volumes of S1 and S2 are equal. Prove this principle.(b) Use Cavalieri's Principle to find the volume of the oblique cylinder shown in the figure.
Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.
A hole of radius is bored through the middle of a cylinder of radius R > r at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.
Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by
Let be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume V of S. Sketch a typical approximating shell. What are its circumference and height? Use shells to find V.
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. (a) y = x4, y = 0, x = 1; about x = 2 (b) y = 4x - x2, y = 3; about x = 1 (c) y = x3, y = 0, x = 1; about y = 1
(1) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (2) Use your calculator to evaluate the integral correct to five decimal places. (a) y = xe-x, y = 0, x = 2; about the y-axis (b) y = cos4x, y = - cos4x, - π/2
Use the Midpoint Rule with n = 5 to estimate the volume obtained by rotating about the y-axis the region under the curve y = √1 + x3, 0 ≤ x ≤ 1.
Each integral represents the volume of a solid. Describe the solid.(a)(b)
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. (a) y = 3√x, y = 0, x = 1 (b) y = e-x2, y = 0, x = 0, x = 1 (c) y = x2, y = 6x - 2x2
Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the -axis the region enclosed by these curves. y = ex, y = √x + 1
Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = sin2 x, y = sin4 x, 0 ≤ x ≤ π; about x = π/2
The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. (a) y = - x2 + 6x - 8, y = 0; about the y-axis (b) y2 - x2 = 1, y = 2; about the x-axis (c) x2 + (y - 1)2 = 1; about the y-axis
Use cylindrical shells to find the volume of the solid. (a) A sphere of radius r (b) A right circular cone with height h and bas radius r
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the -axis. (a) xy = 1, x = 0, y = 1, y = 3 (b) y = x3, y = 8, x = 0 (c) x + y = 3, x = 4 - (y - 1)2
A 360-lb gorilla climbs a tree to a height of 20 ft. Find the work done if the gorilla reaches that height in (a) 10 seconds (b) 5 seconds
A spring has natural length 20 cm. Compare the work W1 done in stretching the spring from 20 cm to 30 cm with the work W2 done in stretching it from 30 cm to 40 cm. How are W2 and W1 related?
Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. (a) A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high. (1) How much work is done in pulling the rope to the top of the building? (2) How much
A tank is full of water. Find the work required to pump the water out of the spout. the fact that water weighs 62.5 lb/ft3.(a)(b)
Suppose that for the tank in Exercise 21 the pump breaks down after 47 Ã— 105 J of work has been done. What is the depth of the water remaining in the tank?
When gas expands in a cylinder with radius r, the pressure at any given time is a function of the volume: P = P(V). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: F = Ï€r2P. Show that the work done by the gas when the volume
(a) Newton's Law of Gravitation states that two bodies with masses m1 and m2 attract each other with a force F = G m1m2 / r2 Where r is the distance between the bodies and G is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from r = a to r = b. (b)
A variable force of 5x-2 pounds moves an object along a straight line when it is x feet from the origin. Calculate the work done in moving the object from x= 1 ft to x = 10 ft.
Shown is the graph of a force function (in newtons) that increases to its maximum value and then remains constant.How much work is done by the force in moving an object a distance of 8 m?
A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length?
Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm. (a) How much work is needed to stretch the spring from 35 cm to 40 cm? (b) How far beyond its natural length will a force of 30 N keep the spring stretched?
Find the average value of the function on the given interval. (a) f(x) = 4x - x2, [0, 4] (b) g(x) = 3√x, [1, 8] (c) f(t) = esin t cos t, [0, π/2]
If f is continuous and ∫31 f(x) dx = 8, show that f takes on the value 4 at least once on the interval [1, 3].
Find the average value of f on [0, 8].
In a certain city the temperature (in oF) hours after 9 AM was modeled by the function T(t) = 50 + 14 sin πt/12 Find the average temperature during the period from 9 AM to 9 PM.
The linear density in a rod 8 m long is 12/√x + 1 kg/m, where x is measured in meters from one end of the rod. Find the average density of the rod.
In Example 1 in Section 3.8 we modeled the world population in the second half of the 20th century by the equation P(t) = 2560e0.017185t. Use this equation to estimate the average world population during this time period.
Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives to the function
(a) Find the average value of f on the given interval. (b) Find such that fave = f(c). (c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f. (1) f(x) = (x - 3)2, [2, 5] (2) f(x) = 2 sin x - sin 2x, [0, π]
Find the area of the region bounded by the given curves. (a) y = x2, y = 4x - x2 (b) y = 1 - 2x2, y = |x| (c) y = sin(πx/2), y = x2 - 2x
Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = cos2x, |x| ≤ π/2, y = 1/4; about x = π/2
Find the volumes of the solids obtained by rotating the region bounded by the curves y = x y = x2 and about the following lines. (a) The x-axis (b) The y-axis (c) y = 2
Let ( be the region bounded by the curves y = tan(x2), x = 1, and y = 0. Use the Midpoint Rule win n = 4 to estimate the following quantities. (a) The area of ( (b) The volume obtained by rotating ( about the x-axis
Each integral represents the volume of a solid. Describe the solid.(a)(b)
The base of a solid is a circular disk with radius 3. Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.
The height of a monument is 20 m. A horizontal cross-section at a distance meters from the top is an equilateral triangle with side 1/4 x meters. Find the volume of the monument.
A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 12 cm to 20 cm?
A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis.(a) If its height is 4 ft and the radius at the top is 4 ft, find the work required to pump the water out of the tank.b) After 4000
If f is a continuous function, what is the limit as h → 0 of the average value of f on the interval [x, x + h]?
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. (a) y = 2x, y = x2; about the x-axis (b) x = 0, x = 9 - y2; about x = - 1 (c) x2 - y2 = a2, x = a + h (where a > 0, h > 0); about the y-axis
(a) Find a positive continuous function f such that the area under the graph of f from 0 to t is A(t) = t2 for all t > 0. (b) A solid is generated by rotating about the -axis the region under the curve y = f(x), where f is a positive function and x ≥0. The volume generated by the part of the
A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The "clock" is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let x = f(x) be continuous on the interval [0, b]
Suppose the graph of a cubic polynomial intersects the parabola y = x2 when x = 0, x = a, and x = b, where 0 < a < b. If the two regions between the curves have the same area, how is b related to a?
If the tangent at a point P on the curve y = x3 intersects the curve again at Q, let A be the area of the region bounded by the curve and the line segment PQ. Let B be the area of the region defined in the same way starting with Q instead of P. What is the relationship between A and B?
The figure shows a horizontal line y = c intersecting the curve y = 8x - 27x3. Find the number c such that the areas of the shaded regions are equal.

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