A trough is filled with a liquid of density 840 kg/m3. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 3e* + 7 sec?x
> Let S be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S. Do you think this method is preferable to slicing? Expl
> Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. 2 3 lim +
> Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. lim 4
> The area labeled B is three times the area labeled A. Express a in terms of b. y. y= e" y= e" B A a
> Suppose h is a function such that h (1) = -2, h'(1) = 2, h"(1) = 3, h (2) = 6, h'(2) = 5, h"(2) = 13, and h" is continuous everywhere. Evaluate f21 h"(u) du.
> A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has e
> (a). If f is continuous, prove that fπ/20 f (cos x) dx = fπ/20 f (sin x) dx (b). Use part (a) to evaluate fπ/20 cos2 x dx and x fπ/20 sin 2x dx.
> If a and b are positive numbers, show that r(1 – x)* dx = [ x*(1 – x)* dx
> If f is continuous on R, prove that fbaf (x + c) dx = fb+ca+c f (x) dx For the case where f (x) > 0, draw a diagram to interpret this equation geometrically as an equality of areas.
> If f is continuous on R, prove that fbaf (-x) dx = f-1-bf (x) dx For the case where f (x) > 0 and 0 < a < b, draw a diagram to interpret this equation geometrically as an equality of areas.
> If f is continuous and f90 f (x) dx = 4, find f30 xf (x2) dx.
> If f is continuous and f40f (x) dx = 10, find f20f (2x) dx.
> The velocity graph of a car accelerating from rest to a speed of 120 km/hover a period of 30 seconds is shown. Estimate the distance traveled during this period. (km/h) 80 40 10 20 30 (seconds)
> Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after weeks is (Notice that production approaches 5000 per week as time goes on, but the initial production is lower b
> Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 L/s. This explains, in part, why the function f (t) = ½ sin (2πt/5) has o
> Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C. C 3r + 1) dx x² + 1
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. 3r u? - 1 du J2x u? + g(x) = Hint: "f(u) du = f(u) du + f* f(u) du /2x
> Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C. 1 C x² + 4 dx X + 2
> Show that f∞0e-x2 dx = f10 √-ln y, dy by interpreting the integrals as areas.
> Show that fπ0x2e-x2 dx = 1/2 f∞0e-x2 dx.
> Estimate the numerical value of f∞0e-x2 dx by writing it as the sum of f40e-x2 dx and f∞4e-x2 dx. Approximate the first integral by using Simpson’s Rule with n = 8 and show that the second integral is smaller than f∞4e-4x dx, which is less than 0.0000001
> Determine how large the number a has to be so that 1 dx < 0.001 x2 + 1
> Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius R the density of
> As we will see in Section 7.4, a radioactive substance decays exponentially: The mass at time t is m (t) = m (0) ekt, where m (0) is the initial mass and is a negative constant. The mean life M of an atom in the substance is For the radioactive carbon
> The figure shows the sun located at the origin and the earth at the point (1, 0). (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU ≈ 1.46 108 km.) There are five locations L1
> The average speed of molecules in an ideal gas is Where M is the molecular weight of the gas, R is the gas constant, T is the gas temperature, and is the molecular speed. Show that 3/2 "p'e-Mu/(2RT) du 4 M 2RT 8RT V TM
> Evaluate the definite integral. Fe4e dx/x√ ln x,
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y = sin't dt - :
> 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 t
> If f∞-∞ f (x) dx is convergent and a and b are real numbers, show that L f(x) dx + [* f(x) dx = [°_f(x) dx + [° f(x) dx
> (a). Show that f∞-∞ x dx is divergent. (b). Show that This shows that we can’t define lim ,xd x dx = 0 Lf(x) dx = lim , f(x) dx
> Evaluate the definite integral. fa0 x√a2 – x2, dx
> 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.
> Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. : p= 2 quarter-circle
> Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. р3 10 y4 (4, 3)
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. у%3 1/х, у— 0, х—D 1, х%3D2
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y = e*, y= 0, x= 0, x= 1
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. Зх + 2у 3 6, у%3D0, х%3D0
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. tan x y = Jo
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y = 4 - x², y = 0
> The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. %3D %3D P.(1, - 2), Р:(3, 4), Р:(-3, —7), Р.(6, — 1)
> The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. т — 6, т — 5, ms — 10; P.(1,5), Р.(3, —2), Р.{(-2, —1)
> Point-masses mi are located on the x-axis as shown. Find the moment M of the system about the origin and the center of mass x. m,= 25 ++ -2 m3= 10 ++++ 7 m2 = 20 ++ 3
> A vertical, irregularly shaped plate is submerged in water. The table shows measurements of its width, taken at the indicated depths. Use Simpson’s Rule to estimate the force of the water against the plate. Depth (m) 2.0 2.5 3.0 3.
> A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate. 2 m water level 12 m 4 m
> Find the volume of the described solid S. The base of S is an elliptical region with boundary curve 9x2 + 4y2 = 36. Cross-sections perpendicular to the -axis are isosceles right triangles with hypotenuse in the base.
> A large tank is designed with ends in the shape of the region between the curves y = 1/2x2 and y = 12, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft with gasoline. (Assume the gasoline’s density is
> Sketch the region in the xy-plane defined by the inequalities x – 2y2 > 0, 1 – x – |y| > 0 and find its area.
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x) = " VT+ r³ dr
> Use cylindrical shells to find the volume of the solid. The solid torus of Exercise 45 in Section 6.2. Exercise 45 in Section 6.2: (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.
> Find the volume of the described solid S. A pyramid with height and rectangular base with dimensions a and 2b.
> Let T be the triangular region with vertices (0, 0), (1, 0), (1, 2), and let V be the volume of the solid generated when T is rotated about the line x = a, where a > 1. Express a in terms of V.
> A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it. -2 m
> A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it. 5 m
> Show that the total length of the ellipse x = a sin θ, y = b cos θ, a > b > 0, is Where e is the eccentricity of the ellipse (e = c/a, where c = √a2 - b2). (T/2 L = 4a Jo VT - e² sin²0 do
> A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density 820 kg/m3 to a depth of 1.5 m. Find (a) the hydrostatic pressure on the bottom of the tank, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of th
> An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find (a) the hydrostatic pressure on the bottom of the aquarium, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the aquarium.
> A steady wind blows a kite due west. The kite’s height above ground from horizontal position x = 0 to x = 80 ft is given by y = 150 – 1/40 (x – 50)2. Find the distance traveled by the kite.
> (a). Newton’s Law of Gravitation states that two bodies with masses m1 and m2 attract each other with a force where is the distance between the bodies and G is the gravitational constant. If one of the bodies is fixed, find the work
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x) = [" arctan t dt 12
> Use Newton’s method to find the coordinates of the inflection point of the curve y = ecosx, 0 < x < π, correct to six decimal places.
> In a steam engine the pressure P and volume V of steam satisfy the equation PV1.4 = k, where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Use Exe
> Use either a CAS or a table of integrals to find the exact length of the curve. y = In(cos x), 0 sxS T/4
> Use either a CAS or a table of integrals to find the exact length of the curve. y? = 4x, 0 < y< 2 %3D
> 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 = sin?x, y = 0, 0 <xS T; about y = -1
> If fave [a, b] denotes the average value of f on the interval [a, b] and a b — с с — а fave[a, c] + b - a fwe[c, b] fave [a, b] - b - a
> Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.3) to the function F (x) = fxaf (t) dt.
> The velocity of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is Where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood (see Example 7 in
> The graph of the concentration function c (t) is shown after a 7-mg injection of dye into a heart. Use Simpson’s Rule to estimate the cardiac output. (mg/L) 6. 4 2 4 6 8 10 1 i (seconds) 14 2.
> The standard deviation for a random variable with probability density function f and mean µ is defined by Find the standard deviation for an exponential density function with mean µ. 11/2 (x-씨)?f(x) dx
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. G(x) = [' cos /F dt
> For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.
> Repeat Exercise 21 for the integral f1-1 √4 – x3, dx. Exercise 21: The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound K for |f (4)(x) | by hand. But computer algebra systems
> Show that the probability density function for a normally distributed random variable has inflection points at x = µ ± σ.
> Use Poiseuille’s Law to calculate the rate of flow in a small human artery where we can take η = 0.027, R = 0.008 cm, t = 2 cm, and P = 4000 dynes/cm2.
> A hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitos is increasing at an estimated rate of 2200 + 10e0.8t per week (where is measured in weeks). By how much does the mosquito population increase betw
> Pareto’s Law of Income states that the number of people with incomes between x = a and x = b is N = fba Ax-k dx, where A and k are constants with A > 0 and k > 1. The average income of these people is Calculate x. ī=÷"Ax!-*
> According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. (a). What is the probability that an adult male chosen at random is between 65 inch
> The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2.5 minutes. (a). Find the probability that a customer has to wait more than 4 minutes. (b). Find the probability that a customer is served wit
> (a). A type of lightbulb is labeled as having an average lifetime of 1000 hours. It’s reasonable to model the probability of failure of these bulbs by an exponential density function with mean µ = 100. Use this model to find the probability that a bulb (
> Show that the median waiting time for a phone call to the company described in Example 4 is about 3.5 minutes.
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. F(x) = [" VI + sec t dt Hint:"V1 + sec t dt = - V1 + sec t dt
> For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. Given the demand curve p = 20 + 1/10x, and the supply curve p
> If a supply curve is modeled by the equation p = 200 + 0.2x3/2, find the producer surplus when the selling price is $400.
> The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound K for |f (4)(x) | by hand. But computer algebra systems have no problem computing and graphing it, so we can easily find a v
> The supply function px (x) for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so ps is an increasing function of x
> A demand curve is given by p = 450/ (x + 8). Find the consumer surplus when the selling price is $10.
> Let f (x) = xe-x if x > 0 and f (x) = 0 for if x < 0. (a). Verify that is a probability density function. (b). Find P (1 < X < 2).
> Let f (x) = 3/64 x √16 – x2, for 0 < x < 4 and f (x) = 0 for all other values of x. (a). Verify that f is a probability density function. (b). Find P (X < 2).
> Let f (t) be the probability density function for the time it takes you to drive to school in the morning, where t is measured in minutes. Express the following probabilities as integrals. (a). The probability that you drive to school in less than 15 min
> Let f (x) be the probability density function for the lifetime of a manufacturer’s highest quality car tire, where is measured in miles. Explain the meaning of each integral. *40,000 (а) 30,000 f(x) dx (b) m f(x) dx 25,000
> A curve is defined by the parametric equations Find the length of the arc of the curve from the origin to the nearest point where there is a vertical tangent line. cos u - du sin u -du y = %3D
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(r) = V? + 4 dx
> 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 that the center of the disk should be positioned at a heig
> Evaluate limx→o = 1/x fx0(1 – tan 2t)1/t, dt.
> Find the point on the parabola y = 1 – x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.
> Estimate the area of the region enclosed by the loop of the curve x = t3 -12t, y = 3t2 + 2t + 5.
> If f is a differentiable function such that f (x) is never 0 and fx0 f (t) dt = [f (x)]2 for all x, find f.
> If f (x) = fx0x2 sin (t2) dt, find f'(x).
