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mathematics
calculus

Questions and Answers of Calculus

The sine integral function is important in electrical engineering. [The integrand f (t) = (sin t) / t is not defined when t = 0, but we know that its limit is 1 when t → 0. So we define f (0) =
Let g(x) f(t) dt, where f is the function whose graph is shown.(a) At what values of do the local maximum and minimum values of occur?(b) Where does attain its absolute maximum value?(c) On what
Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].
A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate f = f (t), where is the time measured in months since its last overhaul. Because a fixed cost A is
A high-tech company purchases a new computing system whose initial value is V. The system will depreciate at the rate f = f (t) and will accumulate maintenance costs at the rate g =g (t), where is
Use a graph to estimate the -intercepts of the curve y = x + x2 – x4. Then use this information to estimate the area of the region that lies under the curve and above the -axis.
Repeat Exercise 41 for the curve y = 2x + 3x4 – 2x6.
If w’ (t) is the rate of growth of a child in pounds per year, what does ∫10 w’ (t) dt represent?
The current in a wire is defined as the derivative of the charge: l (t) = Q’ (t). (See Example 3 in Section 3.3) What does ∫b l (t) dt represent?
If oil leaks from a tank at a rate of r (t) gallons per minute at time t, what does ∫120 r (t) dt represent?
A honeybee population starts with 100 bees and increases at a rate of n’(t) bees per week. What does 1000 + ∫15 n’ (t) dt represent?
In Section 4.8 we defined the marginal revenue function R’(x) as the derivative of the revenue function R(x), where is the number of units sold. What does ∫5000 R’(x) dx represent?
If f(x) is the slope of a trail at a distance of miles from the start of the trail, what does ∫3 f(x) dx represent?
If x is measured in meters and f(x) is measured in newtons, what are the units for ∫100 f(x) dx?
If the units for are feet and the units for a(x) are pounds per foot, what are the units for da/dx? What units does ∫8 a(x) dx have?
The velocity function (in meters per second) is given for a particle moving along a line. Find.(a) The displacement and(b) The distance traveled by the particle during the given time interval.
The acceleration function (in m/s2) and the initial velocity are given for a particle moving along a line. Find(a) The velocity at time and(b) The distance traveled during the given time interval.
The linear density of a rod of length 4 m is given by p(x) = 9 + 2√x measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod.
Water flows from the bottom of a storage tank at a rate of r(t) = 200 – 4t liters per minute, where 0 < t < 50. Find the amount of water that flows from the tank during the first 10 minutes.
The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car.
Suppose that a volcano is erupting and readings of the rate r(t) at which solid materials are spewed into the atmosphere are given in the table. The time is measured in seconds and the units for r(t)
The marginal cost of manufacturing yards of a certain fabric is C’(x) = 3 – 0.01 + 0.000006x2 (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to
Water flows in and out of a storage tank. A graph of the rate of change r(t) of the volume of water in the tank, in liters per day, is shown. If the amount of water in the tank at time t = 0 is
Economists use a cumulative distribution called a Lorenz curve to describe the distribution of income between households in a given country. Typically, a Lorenz curve is defined on [0, 1] with
On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the
Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be(a) Left endpoints and(b) Midpoints. In each case draw a diagram and explain what the Riemann sum
Suppose a particle moves back and forth along a straight line with velocity v(t), measured in feet per second, and acceleration a(t). (a) What is the meaning of ∫60120 v(t) dt? (b) What is
(a) Write ∫5 (x + 2x5) dx as a limit of Riemann sums, taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to compute the limit. (b) Use the
The figure shows the graphs of f, f€™ and ∫x f (t) dt. Identify each graph, and explain your choices.
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its anti-derivative (take C = 0).
Use a graph to give a rough estimate of the area of the region that lies under the curve y = x√ x, 0 < x < 4 . Then find the exact area.
Graph the function f(x) = cos2x sin3 x and use the graph to guess the value of the integral ∫2π f(x) dx. Then evaluate the integral to confirm your guess.
Use the properties of integrals to verify the inequality.
A particle moves along a line with velocity function v(t) = t2 - t, where is measured in meters per second. Find (a) The displacement and (b) The distance traveled by the particle during the time
Let r(t) be the rate at which the world’s oil is consumed, where r is measured in years starting at t = 0 on January 1, 2000, and r(t) is measured in barrels per year. What does ∫3 r(t) dt
A radar gun was used to record the speed of a runner at the times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds.
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
If x sin π x = ∫x2 f(t) dt, where f is a continuous function, find f(4).
(a) Graph several members of the family of functions f(x) = (2cx – x2)/c3 for c > 0 and look at the regions enclosed by these curves and the -axis. Make a conjecture about how the areas of these
The figure shows two regions in the first quadrant: A(t) is the area under the curve y = sin(x2) from to t, B(t) and is the area of the triangle with vertices O, P, and (t, 0). Find lim t→0+
A circular disk of radius is used in an evaporator and is rotated in a vertical plane. If it is to be partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show
The figure shows a region consisting of all points inside a square that are closer to the center than to the sides of the square. Find the area of the region.
For any number fc (x), we let be the smaller of the two numbers (x – c) 2 and (x – c – 2)2. Then we define g(c) = ∫1 fc(x) dx. Find the maximum and minimum values of g(c) if – 2 < c < 2.
Suppose f is continuous, f (0) = 0, f (1) = 1, f’(x) > 0, and ∫1 f(x) dx = 1/3. Find the value of the integral ∫1 f–1 (y) dy.
Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the given curves.
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.
Use a computer algebra system to find the exact area enclosed by the curves y = x5 – 6x3 + 4x and y = x.
Sketch the region in the xy-plane defined by the inequalities x – 2y2 > 0. 1 – x – |y| > 0, and find its area.
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
The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool.
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
The figure shows graphs of the marginal revenue function R€™ and the marginal cost function C€™ for a manufacturer. [Recall from Section 4.8 that R(x) and C(x) represent the
The curve with equation y2 = x2 (x + 3) is called Tschirnhausen cubic. If you graph this curve you will see that part of the curve forms a loop. Find the area enclosed by the loop.
Find the area of the region bounded by the parabola y = x2, the tangent line to this parabola at (1, 1), and the -axis.
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.
(a) Find the number such that the line x = a bisects the area under the curve y = 1/x2, 1 < x < 4(b) Find the number such that the line y = b bisects the area in part (a).
Find the values of such that the area of the region enclosed by the parabolas y = x2 – c2 and y = c2 – x2 is 576.
Suppose that 0 < c < π/2. For what value of is the area of the region enclosed by the curves y = cos x, y = cos (x – c), and x = 0 equal to the area of the region enclosed by the curves y =
For what values of m do the line y = mx and the curve y = x/(x2 + 1) enclose a region? Find the area of the region.
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 line.
Use a graph to find approximate -coordinates of the points of intersection of the given curves. Then find (approximately) the volume of the solid obtained by rotating about the x-axis the region
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.
Each integral represents the volume of a solid. Describe the solid
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
A log 10 m long is cut at 1-meter intervals and its cross-sectional areas (at a distance from the end of the log) are listed in the table. Use the Midpoint Rule with to estimate the volume of the log.
The base of is a circular disk with radius r. Parallel cross-sections perpendicular to the base are isosceles triangles with height and unequal side in the base.(a) Set up an integral for the volume
(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.
Solve Example 9 taking cross-sections to be parallel to the line of intersection of the two planes.
(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
Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.
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 bowl is shaped like a hemisphere with diameter 30 cm. A ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of centimeters. Find the volume of water in the
A hole of radius is bored through 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.
A hole of radius is bored through the center of a sphere of radius > r. Find the volume of the remaining portion of the sphere.
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 1715
Suppose that a region R has area A and lies above the -axis. When R is rotated about the -axis, it sweeps out a solid with volume V1. When R is rotated about the line Y = – K (where is a positive
Let be the solid obtained by rotating the region shown in the figure about the -axis. Explain why it is awkward to use slicing to find the volume V of S. Sketch a typical approximating shell. What
Let S be the solid obtained by rotating the region shown in the figure about the -axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S.
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. Sketch the region and a typical shell.
Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by y = √x and y = x2. Find V both by slicing and by cylindrical shells. In both cases draw a diagram to
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. Sketch the region and a typical shell.
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell.
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.
Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the -axis the region under the curve y = tan x, 0 < x < π/4.
If the region shown in the figure is rotated about the -axis to form a solid, use the Midpoint Rule with n = 5 to estimate the volume of the solid.
Each integral represents the volume of a solid. Describe the solid.
Use a graph to estimate the -coordinates of the points of intersection of the given curves. Then use this information to estimate the volume of the solid obtained by rotating about the
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.
The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.
Use cylindrical shells to find the volume of the solid.
Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h, as
Find the work done in pushing a car a distance of 8 m while 9, exerting a constant force of 900 N.
How much work is done by a weightlifter in raising a 60-kg barbell from the floor to a height of 2 m?
A particle is moved along the -axis by a force that measures 10 / (1 + x)2 pounds at a point feet from the origin. Find the work done in moving the particle from the origin to a distance of 9 ft.
When a particle is located a distance meters from the origin, a force of cos (πx/3) newtons acts on it, how much work is done in moving the particle from x = 1 to x = 2? Interpret your answer by
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?
The table shows values of a force function f(x), where is measured in meters and f(x) in newtons. Use the Midpoint Rule to estimate the work done by the force in moving an object from x = 4 to x = 20.
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?
A spring has a natural length of 20 cm. If a 25-N force is required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 20 cm to 25 cm?
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. How much work is needed to stretch it from 35 cm to 40 cm?

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