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mathematics
calculus

Questions and Answers of Calculus

If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work is needed to stretch it 9 in. beyond its natural length?
12. If 6 J of work is needed to stretch a spring from 10 cm to 12 cm and another 10 J is needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring?
A heavy rope, 50 ft long, weighs 0.5lb/ft and hangs over the edge of a building 120 ft high.(a) How much work is done in pulling the rope to the top of the building?(b) How much work is done in
A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m?
A cable that weighs 2lb/ft is used to lift 800 lb of coal up a mineshaft 500 ft deep. Find the work done.
A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but
A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.8kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant
A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that it’s level with the upper end.
An aquarium 2 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the fact that the density of water is 1000 kg/m)
A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that
A tank is full of water. Find the work required to pump the water out of the outlet. In Exercises 23 and 24 use the fact that water weighs 62.5 lb/ft3.
Suppose that for the tank in Exercise 21 the pump breaks down after 4.7 x 105 J of work has been done. What is the depth of the water remaining in the tank?
Solve Exercise 22 if the tank is half full of oil that has a density of 920 kg/m3.
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
In a steam engine the pressure P and volume V of steam satisfy the equation PV1.4 = k , where is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat
Newton’s Law of Gravitation states that two bodies with masses m1 and m2 attract each other with a force where the distance between the bodies is and G is the gravitational constant. If one of the
Use Newton’s Law of Gravitation to compute the work required to launch a 1000-kg satellite vertically to an orbit 1000 km high. You may assume that Earth’s mass is 5.98 x 1024 kg and is
(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
If f is continuous and ∫3 f(x) dx = 8, show that f takes on the value 4 at least once on the interval [1, 3].
Find the numbers such that the average value of f(x) = 2 + 6x – 3x2 on the interval [0, b] is equal to 3.
The table gives values of a continuous function. Use the Midpoint Rule to estimate the average value of f on [20, 50].
The velocity graph of an accelerating car is shown.(a) Estimate the average velocity of the car during the first 12 seconds.(b) At what time was the instantaneous velocity equal to the average
In a certain city the temperature (in F) hours after 9 A.M. was modeled by the functionFind the average temperature during the period from 9 A.M. to 9 P.M.
If a cup of coffee has temperature 95 C in a room where the temperature is 20 C, then, according to Newton’s Law of Cooling, the temperature of the coffee after minutes is T(t) = 20 + 75e–t/50
The linear density in a rod 8 m long is 12/√x + 1 kg/m, where is measured in meters from one end of the rod. Find the average density of the rod.
If a freely falling body starts from rest, then its displacement is given by s = ½ gt2. Let the velocity after a time T be vr. Show that if we compute the average of the velocities with respect to t
Use the result of Exercise 77 in Section 5.5 to compute the average volume of inhaled air in the lungs in one respiratory cycle.
The velocity of blood that flows in a blood vessel with radius and length at a distance from the central axis is where P is the pressure difference between the ends of the vessel and n is the
Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.2) to the function F(x) = ∫x f (t) dt.
If fave [a, b] denotes the average value of f on the interval [a, b] and a
(a) Draw two typical curves y = f(x) and y = g(x), where f(x) > g(x) for a < x < b. Show how to approximate the area between these curves by a Riemann sum and sketch the corresponding approximating
Suppose that Sue runs faster than Kathy throughout a 1500- meter race. What is the physical meaning of the area between their velocity curves for the first minute of the race?
(a) Suppose is a solid with known cross-sectional areas. Explain how to approximate the volume of by a Riemann sum. Then write an expression for the exact volume.(b) If is a solid of revolution, how
(a) What is the volume of a cylindrical shell?(b) Explain how to use cylindrical shells to find the volume of a solid of revolution.(c) Why might you want to use the shell method instead of slicing?
Suppose that you push a book across a 6-meter-long table by exerting a force f(x) at each point from x = 0 to x = 6. What does ∫6 f(x) dx represent? If f(x) is measured in newtons, what are the
(a) What is the average value of a function f on an interval [a, b]?(b) What does the Mean Value Theorem for Integrals say? What is its geometric interpretation?
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
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.
Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x2 about the following lines:(a) The x-axis (b) The y-axis (c) y = 2
Let R be the region in the first quadrant bounded by the curves y = x3 and y = 2x – x2. Calculate the following quantities.(a) The area of R(b) The volume obtained by rotating R about the x-axis(c)
Let R be the region bounded by the curves y = tan (x2), x = 1, and y = 0. Use the Midpoint Rule with n = 4 to estimate the following.(a) The area of R(b) The volume obtained by rotating R about the
Let R be the region bounded by the curves y = 1 – x2 and y = x6 – x + 1. Estimate the following quantities.(a) The -coordinates of the points of intersection of the curves(b) The area of R (c)
Each integral represents the volume of a solid. Describe the solid.
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
The base of a solid is the region bounded by the parabolas y = x2 and y = 2 – x2. Find the volume of the solid if the cross-sections perpendicular to the -axis are squares with one side lying along
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 x/4 meters. Find the volume of the monument.
(a) The base of a solid is a square with vertices located at (1, 0), (0, 1) (– 1, 0) and (0, – 1). Each cross-section perpendicular to the -axis is a semicircle. Find the volume of the solid.(b)
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 1600-lb elevator is suspended by a 200-ft cable that weighs 10 lb/ft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?
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
Find the average value of the function f (t) = t sin (t2) on the interval [0, 10].
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]?
Let R1 be the region bounded by y = x2, y = 0, and x = b, where b > 0. Let R2 be the region bounded by y = x2, x = 0, and y = b2.(a) Is there a value of such that R1 and R2 have the same area?(b) Is
(a) Find a positive continuous function f such that the area under the graph of f from 0 to is A(t) = t3 for all t > 0.(b) A solid is generated by rotating about the -axis the region under the curve
There is a line through the origin that divides the region bounded by the parabola y = x – x2 and the -axis into two regions with equal area. What is the slope of that line?
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.
A cylindrical glass of radius and height L is filled with water and then tilted until the water remaining in the glass exactly covers its base.(a) Determine a way to €œslice€ the
(a) Show that the volume of a segment of height h of a sphere of radius r is(b) Show that if a sphere of radius 1 is sliced by a plane at a distance x from the center in such a way that the volume of
Archimedes€™ Principle states that the buoyant force on an object partially or fully submerged in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an object
Water in an open bowl evaporates at a rate proportional to the area of the surface of the water. (This means that the rate of decrease of the volume is proportional to the area of the surface.) Show
A sphere of radius 1 overlaps a smaller sphere of radius r in such a way that their intersection is a circle of radius r. (In other words, they intersect in a great circle of the small sphere.) Find
The figure shows a curve C with the property that, for every point P on the middle curve y =2x2, the areas A and B are equal. Find an equation for C.
A paper drinking cup filled with water has the shape of a cone with height and semi vertical angle (see the figure). A ball is placed carefully in the cup, thereby displacing some of the water and
A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The €œclock€ is calibrated for measuring time by placing markings
(a) Determine as a function of w.(b) At what angular speed will the surface of the liquid touch the bottom? At what speed will it spill over the top?(c) Suppose the radius of the container is 2 ft,
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
Suppose we are planning to make a taco from a round tortilla with diameter 8 inches by bending the tortilla so that it is shaped as if it is partially wrapped around a circular cylinder. We will fill
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 the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
Find the average value of f(x) = x2 in x on the interval [1, 3].
A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is , the fuel is consumed at rate r, and the
A particle that moves along a straight line has velocity v(t) = t2 e–t meters per second after seconds. How far will it travel during the first seconds?
We arrived at Formula 6.3.2, V = ∫b 2πx f(x) dx, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its anti-derivative (taking C = 0).
Find the average value of the function f(x) = sin2x cos3x on the interval [– π, π].
Evaluate ∫ sin x cos x dx by four methods: (a) The substitution u = cos x, (b) The substitution u = sin x, (c) The identity sin 2x = 2 sin x cos x, and (d) Integration by parts. Explain
Find the area of the region bounded by the given curves.
Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct.
Find the volume obtained by rotating the region bounded by the given curves about the specified axis.
A particle moves on a straight line with velocity function v(t) = sin wt cos2 wt. Find its position function s = f(t) if f(0) = 0.
Household electricity is supplied in the form of alternating current that varies from 155V to – 155V with a frequency of 60 cycles per second (Hz). The voltage is thus given by the equation E(t) =
Prove the formula, where and are positive integers.
Prove the formula A = ½r2 for the area of a sector of a circle with radius and central angle θ.
Evaluate the integralGraph the integrand and its indefinite integral on the same screen and check that your answer is reasonable.
Use a graph to approximate the roots of the equation x2 √4 – x2 = 2 – x. Then approximate the area bounded by the curve y = x2 √4 – x2 and the line y = 2 – x.
A charged rod of length produces an electric field at point P (a, b) given by where λ is the charge density per unit length on the rod and ε0 is the free space permittivity (see the
Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii r and R. (See the figure.)
A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 ft, what percentage of the total
A torus is generated by rotating the circle x2 + (y – R)2 = r2 about the x-axis. Find the volume enclosed by the torus.
Make a substitution to express the integrand as a rational function and then evaluate the integral.
Use integration by parts, together with the techniques of this section, to evaluate the integral.
Use a graph of f(x) = 1/(x2 – 2x – 3) to decide whether f(x) dx is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find
Graph both y = 1/(x3 – 2x2) and an anti-derivative on the same screen.
Evaluate the integral by completing the square and using Formula 6.
Find the area of the region under the given curve from a to b.
Find the volume of the resulting solid if the region under the curve y = 1/(x2 + 3x + 2) from x = 0 to x = 1 is rotated about (a) The -axis and (b) The -axis.
One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no
Factor x4 + 1 as a difference of squares by first adding and subtracting the same quantity, se this factorization to evaluate f1/(x4 + 1) dx
(a) Use a computer algebra system to find the partial fraction decomposition of the function(b) Use part (a) to find ∫ f(x) dx (by hand) and compare with the result of using the CAS to integrate
(a) Find the partial fraction decomposition of the function(b) Use part (a) to find ∫ f(x) dx and graph f and its indefinite integral on the same screen.(c) Use the graph of f to discover the

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