Find the volume of the described solid S. The base of S is the triangular region with vertices (0, 0), (1, 0), and (0, 1). Cross-sections perpendicular to the y-axis are equilateral triangles.
> The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by c (t) = 20t2-0.6t, 0 < t < 10, where is measured in seconds. Find the cardiac output.
> If n is a positive integer, prove that f10 (ln x) n dx = (-1) n n!
> High blood pressure results from constriction of the arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure. Use Poiseuille’s Law to show that if R0 and P0 are normal values of
> If revenue flows into a company at a rate of f (t) 9000 √1 + 2t, where is measured in years and f (t) is measured in dollars per year, find the total revenue obtained in the first four years.
> If the amount of capital that a company has at time t is f (t), then the derivative, f'(t), is called the net investment flow. Suppose that the net investment flow is √t million dollars per year (where is measured in years). Find the increase in capital
> A movie theater has been charging $7.50 per person and selling about 400 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every 50 cents that they lower the price, the number of moviegoers will increase by 3
> The marginal cost function C (x) was defined to be the derivative of the cost function. (See Sections 3.8 and 4.6.) If the marginal cost of manufacturing meters of a fabric is C'(x) = 5 – 0.008x + 0.000009x2 (measured in dollars per meter) and the fixed
> Use Newton’s method to find the absolute maximum value of the function f (x) = x cos x, 0 < x < π, correct to six decimal places.
> 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?
> 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?
> 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 spr
> Find the average value of the function on the given interval. h(u) — (3 — 2и) -, [-1,1]
> Does the function f (x) = e10|x-2|-x2 have an absolute maximum? If so, find it. What about an absolute minimum?
> Find the average value of the function on the given interval. h(x) — сos'x sin x, [0, п]
> The table shows values of a force function f (x), where is measured in meters and f (x) in newtons. Use Simpson’s Rule to estimate the work done by the force in moving an object a distance of 18 m. 3 9 12 15 18 f(x) 9.8 9.1 8.5 8.0
> 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? FA (N) 30 20 10 1 2 3 4 5 6 7 8 (m)
> Find the average value of the function on the given interval. f(x) = sin 4x, [-7, T]
> Use the result of Exercise 65 in Section 5.5 to compute the average volume of inhaled air in the lungs in one respiratory cycle. Exercise 65 in Section 5.5: Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of
> Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator. x= y + Vy, 1< y< 2
> The graph of f is shown. Evaluate each integral by interpreting it in terms of areas. y= f(x) -2 6 8 4, 2. (b) f(x) dx (c) f, s(x) dx (d) C f(x) dx
> Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. 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?
> 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.
> If a cup of coffee has temperature 950C in a room where the temperature is 200C, then, according to Newton’s Law of Cooling, the temperature of the coffee after t minutes is T (t) = 20 + 75e-t/50. What is the average temperature of the coffee during the
> Show that x2y2 (4 – x2) (4 – y2) < 16 for all numbers and such that |x| < 2 and |y| < 2.
> Investigate the family of functions f (x) = =cxe – cx2. What happens to the maximum and minimum points and the inflection points as changes? Illustrate your conclusions by graphing several members of the family.
> In a certain city the temperature (in 0F) t hours after 9 AM was modeled by the function Find the average temperature during the period from 9 AM to 9 PM. T(t) = 50 + 14 sin 12
> 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 velocity? v. (km/h) 60 40 20 4 8 12 1 (seconds)
> The table gives values of a continuous function. Use Simpson’s Rule to estimate the average value of f on [20, 50]. 20 25 30 35 40 45 50 f(x) 42 38 31 29 35 48 60
> Find the numbers such that the average value of f (x) = 2 +6x – 3x2 on the interval [0, b] is equal to 3.
> If f is continuous and f31f(x) dx = 8, show that f takes on the value 4 at least once on the interval [1, 3].
> 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 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.
> f (x) = 2 sin x – sin 2x, [0, π] (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.
> Of the infinitely many lines that are tangent to the curve y = -sin x and pass through the origin, there is one that has the largest slope. Use Newton’s method to find the slope of that line correct to six decimal places.
> f (x) = ln x, [1, 3] (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.
> 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 ahe
> f (x) = (x – 3)2, [2, 5] (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.
> Find the values of p for which the integral converges and evaluate the integral for those values of p. 1 dx x(In x)" Je
> The integral f∞01/√x(1 + x) is improper for two reasons: The interval [0, ∞] is infinite and the integrand has an infinite discontinuity at 0. Evaluate it by expressing it as a sum of improper integra
> Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.
> (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 Prin
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. - arctan x 2 + e*
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. x + 1 dx x4 - x
> The base of S is a circular disk with radius r. Parallel crosssections perpendicular to the base are isosceles triangles with height h and unequal side in the base. (a). Set up an integral for the volume of S. (b). By interpreting the integral as an area
> Find the volume of the described solid S. The base of S is the same base as in Exercise 42, but crosssections perpendicular to the x-axis are isosceles triangles with height equal to the base. Exercise 42: The base of is the region enclosed by the par
> Find the approximations Tn, Mn, and Sn for n = 6 and 12. Then compute the corresponding errors ET, EM, and ES. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? I
> Show that, for all positive values of x and y, ex+y /xy > e2.
> Find the volume of the described solid S. The base of is the region enclosed by the parabola y = 1 – x2 and the x-axis. Cross-sections perpendicular to the -axis are squares.
> Find the volume of the described solid S. The base of S is the same base as in Exercise 40, but crosssections perpendicular to the x-axis are squares. Exercise 40: The base of S is the triangular region with vertices (0, 0), (1, 0), and (0, 1). Cross-s
> Find the average value of the function on the given interval. f(@) = sec²(e/2), [0, 7/2]
> Find the volume of the described solid S. The base of S is a circular disk with radius r. Parallel crosssections perpendicular to the base are squares
> Use cylindrical shells to find the volume of the solid. A right circular cone with height h and base radius r
> The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. у 3 5, у — х+ (4/х); about x — -1
> The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x = (y – 3)', x = 4; about y = 1
> Find the total length of the astroid x = a cos3θ, y = sin3θ, where a > 0.
> Find the average value of the function on the given interval. g(x) = x, [1,8]
> Find the interval [a, b] for which the value of the integral fba (2 + x – x2) dx is a maximum.
> Let I = f40 f(x) dx, where f is the function whose graph is shown. (a). Use the graph to find L2, R2, and M2. (b). Are these underestimates or overestimates of I? (c). Use the graph to find T2. How does it compare with I? (d). For any value of n, list
> The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. y = -x² + 6x – 8, y = 0; about the y-axis
> A hawk flying at 15m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation y = 180 – x2/45 until it hits the ground, where is its height above the ground and is the horizontal dis
> Use either a CAS or a table of integrals to find the exact length of the curve. y = In x, 1<xs3
> f (x) = 2x/ (1 + x2)2, [0, 2] (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.
> Find the average value of the function on the given interval. fa) — 4х — х, [О, 4]
> Find the exact length of the curve. x = y3/2, 0 < y < 1
> Find the exact length of the curve. y2 = 4 (x + 4)3, 0 < x < 2, y > 0
> Find the exact length of the curve. x = 1 + 3t2, y = 4 + 2t3, 0 < t < 1
> Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. x = t cos t, y =t sin t, 0 st< 27 0 <t< 27
> 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. 2. 2. 2.
> Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. x =t + cos t, y=t- sin t, 0st< 2m
> Let f (x) = 0 if is any rational number and f (x) = 1 if is any irrational number. Show that f is not integrable on [0, 1].
> Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. x= y? – 2y, 0 <y<2
> Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. y = sin x, 0<x<T
> If the region shown in the figure is rotated about the y-axis to form a solid, use Simpson’s Rule with n = 10 to estimate the volume of the solid. y 5 4 1 1 2 3 4 5 6 7 8 9 10 11 12 í 3. 2.
> Use Simpson’s Rule with n = 10 to estimate the volume obtained by rotating about the y-axis the region under the curve y = √1 + x3, 0 < x < 1.
> Find the length of the loop of the curve x = 3t – t3, y = 3t2.
> (a). In Example 2 in Section 1.7 we showed that the parametric equations x = cos t, y = sin t, 0 < t < 2π, represent the unit circle. Use these equations to show that the length of the unit circle has the expected value. (b). In Example 3 in Section 1.7
> Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator. x = sin t, y = 1?, 0 st< 2n
> Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator. y = xe, 0<I<5
> Sketch the region in the plane consisting of all points (x, y) such that 2xy < |x – y|< x? + y?
> Graph the curve and find its exact length. x = et + e-t, y = 5 – 2t, 0 < t < 3
> Graph the curve and find its exact length. x = et cos t, y = et sin t, 0 < t < π
> 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 velocity data for the shuttle between liftoff and the je
> Graph the curve and find its exact length. y = x3/3 + 1/4x, 1 < x < 2
> Graph the curve and find its exact length. x = et – t, y = 4et/2, -8 < t < 3
> Find the exact length of the curve. x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ), 0 < θ < π
> Find the exact length of the curve. y = 1/4x2 – 1/2 lnx, 1 < x < 2
> Find the exact length of the curve. y = √x – x2 + sin-1 (√x)
> Use the arc length formula (2) to find the length of the curve y – 2x - 5, -1 < x < 3. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.
> 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 x-axis. х%3D1+y;, х30, у%31, у%3D2
> A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cm/s into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely
> 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 explain your method.
> 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. y = 4(x – 2), y =x² – 4x + 7
> 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. y = 3 + 2x – x², x+ y = 3
> 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 endpoints (0, 0) and (1, 1), and is continuous, increasing, a
> 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. y = e*, y = 0, x= 0, x= 1
> 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. y = x², y = 0, x= 1
> The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x² + (y – 1)? = 1; about the y-axis
> The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. y = -x? + 6x – 8, y = 0; about the x-axis
> 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. у%3D 1/х, у—0, х— 1, х—2
> 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 = x'sin x, y = 0, 0 <x< T; about x = -1
