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study help
mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
Use the substitution u = x4 + 1 to evaluate the integral [x²√x + x7 √x4 + 1 dx.
Diastolic blood pressure in adults is normally distributed with μ = 80 mm Hg and σ = 12 mm Hg. In a random sample of 300 adults, how many would be expected to have a diastolic blood pressure below 70 mm Hg?
Solve the initial value problems for y as a function of x. X dy dx = √x² - 4₁ x ≥ 2, y(2) = 0 4,
Serum albumin in healthy 20-year-old males is normally distributed with μ = 4.4 and σ = 0.2. How likely is it for a healthy 20-year-old male to have a level in the range 4.3 to 4.45?
Heat capacity Cν is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal / deg-mol (calories per degree gram molecular weight). The heat capacity of oxygen depends on its temperature T and satisfies the formulaUse
Use reduction formulas to evaluate the integrals. [₁ 16x³(In x)² dx
An automobile computer gives a digital readout of fuel consumption in gallons per hour. During a trip, a passenger recorded the fuel consumption every 5 min for a full hour of travel.a. Use the Trapezoidal Rule to approximate the total fuel consumption during the hour.b. If the automobile covered
Solve the initial value problems for y as a function of x. √x²-9 dx = 1, x > 3, y(5) = In 3
Evaluate the integrals. Some integrals do not require integration by parts. 1/V2 2x sin¹ (x²) dx
Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. Jese sec³ (et - 1) dt
Solve the initial value problems for y as a function of x. (x² + 4) dy dx 3, y(2) = 0
Evaluate the integrals. Some integrals do not require integration by parts. [x x tan-¹x dx
A manufacturer of generator shafts finds that it needs to add additional weight to its shafts in order to achieve proper static and dynamic balance. Based on experimental tests, the average weight it needs to add is μ = 35 gm with σ = 9 gm. Assuming a normal distribution, from 1000 randomly
To meet the demand for parking, your town has allocated the area shown here. As the town engineer, you have been asked by the town council to find out if the lot can be built for $11,000. The cost to clear the land will be $0.10 a square foot, and the lot will cost $2.00 a square foot to pave. Use
Evaluate the integrals. Some integrals do not require integration by parts. 1.x² X x² tan -1½ dx
Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. csc ³ Ve Vo 0 f= do
Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. So 0 2√x² + 1 dx
A taxicab company in New York City analyzed the daily number of miles driven by each of its drivers. It found the average distance was 200 mi with a standard deviation of 30 mi. Assuming a normal distribution, what prediction can we make about the percentage of drivers who will log in either more
The germination rate of a particular seed is the percentage of seeds in the batch which successfully emerge as plants. Assume that the germination rate for a batch of sunflower seeds is 80%, and that among a large population of n seeds the number of successful germinations is normally distributed
Find the area of the region enclosed by the curve y = x cos x and the x-axis fora. π/2 ≤ x ≤ 3π/2.b. 3π/2 ≤ x ≤ 5π/2.c. 5π/2 ≤ x ≤ 7π/2.d. What pattern do you see? What is the area between the curve and the x-axis forn an arbitrary positive integer? Give reasons for your answer.
Solve the initial value problems for x as a function of t. (1²+2t) dx dt = 2x + 2 (t, x > 0), x(1) = 1
Find the area of the region in the first quadrant that is enclosed by the coordinate axes and the curve y = V9 - x²/3.
Find the area of the region enclosed by the curve y = x sin x and the x-axis fora. 0 ≤ x ≤ π.b. π ≤ x ≤ 2π.c. 2π ≤ x ≤ 3π.d. What pattern do you see here? What is the area between the curve and the x-axis for nπ ≤ x ≤ (n + 1)π, n an arbitrary nonnegative integer? Give reasons
A fair coin is tossed four times and the random variable X assigns the number of tails that appear in each outcome.a. Determine the set of possible outcomes.b. Find the value of X for each outcome.c. Create a probability bar graph for X, as in Figure 8.21. What is the probability that at least two
Solve the initial value problems for x as a function of t. dx (t+1) = dt x² + 1 (t-1), x(0) = 0
Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. √3/2 dy (1 - y²)5/2
Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. 1 2 (r² - 1)³/2 dr r al
Find the volume of the solid generated by revolving the shaded region about the indicated axis.The x-axis y 2- y = (0.5, 2.68) 0.5 3 √3x - x² (2.5, 2.68) 2.5 X
Suppose you toss a fair coin n times and record the number of heads that land. Assume that n is large and approximate the discrete random variable X with a continuous random variable that is normally distributed with μ = n/2 and σ = √n/2. If n = 400, find the given probabilities.a. P(190 ≤ X
Find the volume of the solid generated by revolving the shaded region about the indicated axis.The y-axis 1 y 0 y = 2 (x + 1)(2 - x) 1
Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. 0 1/√3 dt (t² + 1) 7/2
Evaluatea. Integration by parts.b. A u-substitution.c. A trigonometric substitution. Sin xp zx - I^exf
Suppose that a boat is positioned at the origin with a water skier tethered to the boat at the point (30, 0) on a rope 30 ft long. As the boat travels along the positive y-axis, the skier is pulled behind the boat along an unknown path y = ƒ(x), as shown in the accompanying figure.a. Show thatb.
Consider the region bounded by the graphs of y = sin-1 x, y = 0, and x = 1/2.a. Find the area of the region.b. Find the centroid of the region.
Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = ex, and the line x = ln 2 about the line x = ln 2.
Require the use of various trigonometric identities before you evaluate the integrals. Isin sin² 0 cos 30 de
Three people are asked their opinion in a poll about a particular brand of a common product found in grocery stores. They can answer in one of three ways: “Like the product brand” (L), “Dislike the product brand” (D), or “Undecided” (U). For each outcome, the random variable X assigns
Find the x-coordinate of the centroid of this region to two decimal places. y (3, 1.83) от 3 = 4x² + 13x - 9 x3 + 2x2 - 3x (5, 0.98) и 5 X
Sociologists sometimes use the phrase “social diffusion” to describe the way information spreads through a population. The information might be a rumor, a cultural fad, or news about a technical innovation. In a sufficiently large population, the number of people x who have the information is
Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e-x, and the line x = 1a. About the y-axis.b. About the line x = 1.
Require the use of various trigonometric identities before you evaluate the integrals. [ cos² 20 sin 0 de
A component of a spacecraft has both a main system and a backup system operating throughout a flight. The probability that both systems fail sometime during the flight is 0.0148. Assuming that each system separately has the same failure rate, what is the probability that the main system fails
Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = tan-1 x, and the line x = √3.
Require the use of various trigonometric identities before you evaluate the integrals. [ cos³ € sin 20 de
Find the centroid of the region cut from the first quadrant by the curveand the line x = 3. y = 1/√x + 1
Many chemical reactions are the result of the interaction of two molecules that undergo a change to produce a new product. The rate of the reaction typically depends on the concentrations of the two kinds of molecules. If a is the amount of substance A and b is the amount of substance B at time t =
Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos x, 0 ≤ x ≤ π/2, abouta. The y-axis.b. The line x = π/2.
A retarding force, symbolized by the dashpot in the accompanying figure, slows the motion of the weighted spring so that the mass’s position at time t isFind the average value of y over the interval 0 ≤ t ≤ 2π. y = 2e¹ cost, t≥ 0.
Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin x, 0 ≤ x ≤ π, abouta. The y-axis.b. The line x = π.
Require the use of various trigonometric identities before you evaluate the integrals. [sin sin³ 0 cos 20 de
In a mass-spring-dashpot system like the one in Exercise 61, the mass’s position at time t isFind the average value of y over the interval 0 ≤ t ≤ 2π.Exercise 61A retarding force, symbolized by the dashpot in the accompanying figure, slows the motion of the weighted spring so that the
Consider the region bounded by the graphs of y = ln x, y = 0, and x = e.a. Find the area of the region.b. Find the volume of the solid formed by revolving this region about the x-axis.c. Find the volume of the solid formed by revolving this region about the line x = -2.d. Find the centroid of the
The head of your firm’s accounting department has asked you to find a formula she can use in a computer program to calculate the year-end inventory of gasoline in the company’s tanks. A typical tank is shaped like a right circular cylinder of radius r and length L, mounted horizontally, as
Require the use of various trigonometric identities before you evaluate the integrals. [si sin cos cos 30 de
Require the use of various trigonometric identities before you evaluate the integrals. [sin sin sin 20 sin 30 de
A thin plate of constant density δ = 1 occupies the region enclosed by the curve y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.
Consider the region bounded by the graphs of y = tan-1 x, y = 0, and x = 1.a. Find the area of the region.b. Find the volume of the solid formed by revolving this region about the y-axis.
Use a CAS to perform the integrations.Evaluate the integrals.a.b.c.d. What pattern do you see? Predict the formula forand then see if you are correct by evaluating it with a CAS.e. What is the formula forCheck your answer using a CAS. In x z.X dx
Use any method to evaluate the integral. sec³ x tan x dx
Use a CAS to perform the integrations.Evaluate the integralsa.b.c.d. What pattern do you see? Predict the formula for ∫x4 ln xdx and then see if you are correct by evaluating it with a CAS.e. What is the formula for ∫xn ln xdx, n ≥ 1? Check your answer using a CAS. fx x ln x dx
Use the integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve y = x2, -1 ≤ x ≤ 1, about the x-axis.
Use integration by parts to establish the reduction formula. Tx x" cos x dx = xn sin x - nf x² xn-1 sin x dx
Use any method to evaluate the integral. sin³ x cos* x dx
Use integration by parts to establish the reduction formula. [x² xn sin x dx = -x" cos x + nxn-1 cos x dx - nf x²-1
Show that ∫∞-∞ ƒ(x) dx may not equal limb→∞ ∫b-b ƒ(x) dx diverges and hence thatdiverges. Then show that 0 2x dx x² + 1 X
Find the values of p for which each integral converges.a.b. 2 S 1 dx x(ln x)P
Use integration by parts to establish the reduction formula. xneax n [x²e" dx = x²6²-fx²-¹6 xneax a xn-leax dx, a 0
Use any method to evaluate the integral. tan² x J CSC X dx
Use any method to evaluate the integral. J cot. cos²x dx
Use integration by parts to establish the reduction formula. Ja (In x)" dx = x(In x)" - nf (In (In x)-1 dx
a. Use a CAS to evaluatewhere n is an arbitrary positive integer. Does your CAS find the result?b. In succession, find the integral when n = 1, 2, 3, 5, and 7. Comment on the complexity of the results.c. Now substitute x = (π/2) - u and add the new and old integrals. What is the value ofThis
Use any method to evaluate the integral. Tx x sin² x dx
Use integration by parts to establish the reduction formula. [x J.xm xm(In x) dx = xm+1 m + 1 (In x)n xm (In x)"-1 dx, m = -1 - n m + 1
Use Example 5 to show that Cπ/2 1.™ sin 0 sin" x dx π/2 - for = cos" x dx EIN 1.3.5 (n-1) 2.4.6...n 2 2.4.6. (n - 1) 1.3.5...n n even n odd
Use any method to evaluate the integral. fre x cos³ x dx
The infinite region in the first quadrant between the curve y = e-x and the x-axis.Find the area of the region.
The infinite region in the first quadrant between the curve y = e-x and the x-axis.Find the centroid of the region.
Integration by parts leads to a rule for integrating inverses that usually gives good results:The idea is to take the most complicated part of the integral, in this case ƒ -1(x), and simplify it first. For the integral of ln x, we getFor the integral of cos-1 x we getUse the formulato evaluate the
Integration by parts leads to a rule for integrating inverses that usually gives good results:The idea is to take the most complicated part of the integral, in this case ƒ -1(x), and simplify it first. For the integral of ln x, we getFor the integral of cos-1 x we getUse the formulato evaluate the
Integration by parts leads to a rule for integrating inverses that usually gives good results:The idea is to take the most complicated part of the integral, in this case ƒ -1(x), and simplify it first. For the integral of ln x, we getFor the integral of cos-1 x we getUse the formulato evaluate the
Integration by parts leads to a rule for integrating inverses that usually gives good results:The idea is to take the most complicated part of the integral, in this case ƒ -1(x), and simplify it first. For the integral of ln x, we getFor the integral of cos-1 x we getUse the formulato evaluate the
Another way to integrate ƒ-1(x) (when ƒ-1 is integrable, of course) is to use integration by parts with u = ƒ-1(x) and dν = dx to rewrite the integral of ƒ-1 ascompare the results of using Equations (4) and (5).Equations (4) and (5) give different formulas for the integral of cos-1 x:a.b.
Another way to integrate ƒ-1(x) (when ƒ-1 is integrable, of course) is to use integration by parts with u = ƒ-1(x) and dν = dx to rewrite the integral of ƒ-1 ascompare the results of using Equations (4) and (5).Equations (4) and (5) lead to different formulas for the integral of tan-1
Use integration by parts to obtain the formula | - [ V₁ = x² dx = 1/2 x VI = x² + 1/ / 2 1 V1- √T-1² dx.
Evaluate the integrals with (a) Eq. (4) and (b) Eq. (5). In each case, check your work by differentiating your answer with respect to x. S sinh 1 x dx
Evaluate the integrals with (a) Eq. (4) and (b) Eq. (5). In each case, check your work by differentiating your answer with respect to x. Stanhi tanh-1 x dx
The infinite region in the first quadrant between the curve y = e-x and the x-axis.Find the volume of the solid generated by revolving the region about the y-axis.z
Find the center of gravity of the region bounded by the x-axis, the curve y = sec x, and the lines x = -π/4, x = π/4.
The infinite region in the first quadrant between the curve y = e-x and the x-axis.Find the volume of the solid generated by revolving the region about the x-axis.
Use a CAS to explore the integrals for various values of p (include noninteger values). For what values of p does the integral converge? What is the value of the integral when it does converge? Plot the integrand for various values of p. 0 XP xP In x dx
Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.
Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.
Use a CAS to explore the integrals for various values of p (include noninteger values). For what values of p does the integral converge? What is the value of the integral when it does converge? Plot the integrand for various values of p. -∞ xp |x| UI dx
Find the area between the x-axis and the curve y = √1 + cos 4x, 0 ≤ x ≤ π.
The region in Exercise 71 is revolved about the x-axis to generate a solid.a. Find the volume of the solid.b. Show that the inner and outer surfaces of the solid have infinite area.Exercise 71Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.
Use a CAS to explore the integrals for various values of p (include noninteger values). For what values of p does the integral converge? What is the value of the integral when it does converge? Plot the integrand for various values of p. e 80 xP In x dx
Use a CAS to explore the integrals for various values of p (include noninteger values). For what values of p does the integral converge? What is the value of the integral when it does converge? Plot the integrand for various values of p. xp x UI dx 0 J
Find the centroid of the region bounded by the graphs of y = x + cos x and y = 0 for 0 ≤ x ≤ 2π.
Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.
Use a CAS to evaluate the integrals. 0 2/TT sin = dx X
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