Exercises 16 refer to the region R in the first quadrant enclosed by the x-axis and the graph of the function y = 4x - x
Read moreTextbook Solutions for Calculus: Graphical, Numerical, Algebraic
Chapter 5 Problem 56
Question
A town wants to drain and fill the small polluted swamp shown below. The swamp averages 5 ft deep. About how many cubic yards of dirt will it take to fill the area after the swamp is drained?
Solution
The first step in solving 5 problem number 56 trying to solve the problem we have to refer to the textbook question: A town wants to drain and fill the small polluted swamp shown below. The swamp averages 5 ft deep. About how many cubic yards of dirt will it take to fill the area after the swamp is drained?
From the textbook chapter The Definite Integral you will find a few key concepts needed to solve this.
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Title
Calculus: Graphical, Numerical, Algebraic 3
Author
Ross L. Finney
ISBN
9780132014083
A town wants to drain and fill the small polluted swamp
Chapter 5 textbook questions
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Chapter 5: Problem 1 Calculus: Graphical, Numerical, Algebraic 3
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Chapter 5: Problem 2 Calculus: Graphical, Numerical, Algebraic 3
Exercises 16 refer to the region R in the first quadrant enclosed by the x-axis and the graph of the function y = 4x - x
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Chapter 5: Problem 3 Calculus: Graphical, Numerical, Algebraic 3
Exercises 16 refer to the region R in the first quadrant enclosed by the x-axis and the graph of the function y = 4x - x
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Chapter 5: Problem 4 Calculus: Graphical, Numerical, Algebraic 3
Exercises 16 refer to the region R in the first quadrant enclosed by the x-axis and the graph of the function y = 4x - x
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Chapter 5: Problem 5 Calculus: Graphical, Numerical, Algebraic 3
Exercises 16 refer to the region R in the first quadrant enclosed by the x-axis and the graph of the function y = 4x - x
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Chapter 5: Problem 6 Calculus: Graphical, Numerical, Algebraic 3
Exercises 16 refer to the region R in the first quadrant enclosed by the x-axis and the graph of the function y = 4x - x
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Chapter 5: Problem 7 Calculus: Graphical, Numerical, Algebraic 3
Use a calculator program to compute the RAM approximations in the following table for the area under the graph of y = 1/x from x = 1 to x = 5.
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Chapter 5: Problem 8 Calculus: Graphical, Numerical, Algebraic 3
(Continuation of Exercise 7) Use the Fundamental Theorem of Calculus to determine the value to which the sums in the table are converging.
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Chapter 5: Problem 9 Calculus: Graphical, Numerical, Algebraic 3
Suppose ! 2 !2 f "x# dx " 4, ! 5 2 f "x# dx " 3, ! 5 !2 g"x# dx " 2. Which of the following statements are true, and which, if any, are false? (a) ! 2 5 f "x# dx " !3 True (b) ! 5 !2 $ f "x# # g"x#% dx " 9 True (c) f "x# $ g"x# on the interval !2 $ x $ 5
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Chapter 5: Problem 10 Calculus: Graphical, Numerical, Algebraic 3
The region under one arch of the curve y " sin x is revolved around the x-axis to form a solid. (a) Use the method of Example 3, Section 5.1, to set up a Riemann sum that approximates the volume of the solid. (b) Find the volume using NINT.
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Chapter 5: Problem 11 Calculus: Graphical, Numerical, Algebraic 3
The accompanying graph shows the velocity "m%sec# of a body moving along the s-axis during the time interval from t " 0 to t " 10 sec. (a) About how far did the body travel during those 10 seconds? 26.5 m (b) Sketch a graph of position (s)
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Chapter 5: Problem 12 Calculus: Graphical, Numerical, Algebraic 3
The interval $0, 10% is partitioned into n subintervals of length Dx " 10%n. We form the following Riemann sums, choosing each ck in the kth subinterval. Write the limit as n of each Riemann sum as a definite integral.
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Chapter 5: Problem 13 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 13 and 14, find the total area between the curve and the x-axis
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Chapter 5: Problem 14 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 13 and 14, find the total area between the curve and the x-axis
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Chapter 5: Problem 15 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 16 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 17 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 18 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 19 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 20 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 21 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 22 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 23 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 24 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 1524, evaluate the integral analytically by using the Integral Evaluation Theorem (Part 2 of the Fundamental Theorem, Theorem 4).
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Chapter 5: Problem 25 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 2529, evaluate the integral.
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Chapter 5: Problem 26 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 2529, evaluate the integral.
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Chapter 5: Problem 27 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 2529, evaluate the integral.
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Chapter 5: Problem 28 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 2529, evaluate the integral.
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Chapter 5: Problem 29 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 2529, evaluate the integral.
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Chapter 5: Problem 30 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 3032, evaluate the integral by interpreting it as area and using formulas from geometry.
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Chapter 5: Problem 31 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 3032, evaluate the integral by interpreting it as area and using formulas from geometry.
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Chapter 5: Problem 32 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 3032, evaluate the integral by interpreting it as area and using formulas from geometry.
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Chapter 5: Problem 33 Calculus: Graphical, Numerical, Algebraic 3
A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced.
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Chapter 5: Problem 34 Calculus: Graphical, Numerical, Algebraic 3
A sled powered by a wound rubber band moves along a track until friction and the unwinding of the rubber band gradually slow it to a stop. A speedometer in the sled monitors its speed, which is recorded at 3-second intervals during the 27-second run
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Chapter 5: Problem 35 Calculus: Graphical, Numerical, Algebraic 3
Your friend knows how to compute integrals but never could understand what difference the dx makes, claiming that it is irrelevant. How would you explain to your friend why it is necessary?
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Chapter 5: Problem 36 Calculus: Graphical, Numerical, Algebraic 3
The function is discontinuous at 0, but integrable on !"4, 4". Find !4 "4 f#x$ dx
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Chapter 5: Problem 37 Calculus: Graphical, Numerical, Algebraic 3
Show that 0 # !1 0 %1&&$&sin&&2 x& dx # %2&.
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Chapter 5: Problem 38 Calculus: Graphical, Numerical, Algebraic 3
Find the average value of (a) y ! %x& over the interval !0, 4". % 4 3 % (b) y ! a%x& over the interval !0, a".
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Chapter 5: Problem 39 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 3942, find dy/dx
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Chapter 5: Problem 40 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 3942, find dy/dx
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Chapter 5: Problem 41 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 3942, find dy/dx
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Chapter 5: Problem 42 Calculus: Graphical, Numerical, Algebraic 3
In Exercises 3942, find dy/dx
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Chapter 5: Problem 43 Calculus: Graphical, Numerical, Algebraic 3
Including start-up costs, it costs a printer $50 to print 25 copies of a newsletter, after which the marginal cost at x copies is % d d c x % ! % % 2 &x % dollars per copy. Find the total cost of printing 2500 newsletters. $
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Chapter 5: Problem 44 Calculus: Graphical, Numerical, Algebraic 3
Rich Wholesale Foods, a manufacturer of cookies, stores its cases of cookies in an airconditioned warehouse for shipment every 14 days. Rich tries to keep 600 cases on reserve to meet occasional peaks in demand, so a typical 14-day inventory function is I#t$ ! 600 $ 600t, 0 # t # 14. The holding cost for each case is 4 per day. Find Richs average daily inventory and average daily holding cost (that is, the average of I(x) for the 14-day period, and this average multiplied by the holding cost).
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Chapter 5: Problem 45 Calculus: Graphical, Numerical, Algebraic 3
Solve for x: !x 0 #t 3 " 2t $ 3$ dt ! 4
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Chapter 5: Problem 46 Calculus: Graphical, Numerical, Algebraic 3
Suppose f #x$ has a positive derivative for all values of x and that f #1$ ! 0. Which of the following statements must be true of g#x$ ! ' x 0 f #t$ dt? (a) g is a differentiable function of x. True (b) g is a continuous function of x. True (c) The graph of g has a horizontal tangent line at x ! 1. True (d) g has a local maximum at x ! 1. False (e) g has a local minimum at x ! 1. True (f) The graph of g has an inflection point at x ! 1. False (g) The graph of dg&dx crosses the x-axis at x ! 1.
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Chapter 5: Problem 47 Calculus: Graphical, Numerical, Algebraic 3
Suppose F#x$ is an antiderivative of f #x$ ! %1&&$&x 4 &. Express !1 0%1&&$&x&4 dx in terms of F.
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Chapter 5: Problem 48 Calculus: Graphical, Numerical, Algebraic 3
Express the function y#x$ with % d d y x % ! % sin x x % and y#5$ ! 3 as a definite integral
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Chapter 5: Problem 49 Calculus: Graphical, Numerical, Algebraic 3
Show that y ! x2 $ !x 1 1&t dt $ 1 satisfies both of the following conditions: i. y ' ! 2 " %x 1 %2 ii. y ! 2 and y( ! 3 when x ! 1
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Chapter 5: Problem 50 Calculus: Graphical, Numerical, Algebraic 3
Which of the following is the graph of the function whose derivative is dy&dx ! 2x and whose value at x ! 1 is 4? Explain your answer
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Chapter 5: Problem 51 Calculus: Graphical, Numerical, Algebraic 3
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 minutes for a full hour of travel
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Chapter 5: Problem 52 Calculus: Graphical, Numerical, Algebraic 3
Skydivers A and B are in a helicopter hovering at 6400 feet. Skydiver A jumps and descends for 4 sec before opening her parachute. The helicopter then climbs to 7000 feet and hovers there. Forty-five seconds after A leaves the aircraft, B jumps and descends for 13 sec before opening her parachute. Both skydivers descend at 16 ft!sec with parachutes open. Assume that the skydivers fall freely (with acceleration "32 ft!sec2) before their parachutes open. (a) At what altitude does As parachute open? 6,144 ft (b) At what altitude does Bs parachute open? 4,296 ft (c) Which skydiver lands first?
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Chapter 5: Problem 53 Calculus: Graphical, Numerical, Algebraic 3
The figure below shows an interval of length 2h with a trapezoid, a midpoint rectangle, and a parabolic region on it. (a) Show that the area of the trapezoid plus twice the area of the rectangle equals h(y1 # y3) # 2(2hy2) $ h(y1 # 4y2 # y3) h! y1 # 4y2 # y3". (b) Use the result in part (a) to prove that S2n $% 2 MRA 3 M %n #
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Chapter 5: Problem 54 Calculus: Graphical, Numerical, Algebraic 3
The graph of a function f consists of a semicircle and two line segments as shown below. Let g!x" $ !x 1 f !t" dt. (a) Find g!1". 0 (b) Find g!3". "1 (c) Find g!"1". "p (d) Find all values of x on the open interval !"3, 4" at which g has a relative maximum. x $ 1 (e) Write an equation for the line tangent to the graph of g at x $ "1. (f) Find the x-coordinate of each point of inflection of the graph of g on the open interval !"3, 4". x $ "1, x $ 2 (g) Find the range of g.
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Chapter 5: Problem 55 Calculus: Graphical, Numerical, Algebraic 3
What is the total area under the curve y $ e"x 2!2? The graph approaches the x-axis as an asymptote both to the left and the right, but quickly enough so that the total area is a finite number. In fact, NINT !e"x 2!2, x, "10, 10" computes all but a negligible amount of the area. (a) Find this number on your calculator. Verify that NINT !e"x 2!2, x, "20, 20" does not increase the number enough for the calculator to distinguish the difference. See above. (b) This area has an interesting relationship to !. Perform various (simple) algebraic operations on the number to discover what it is.
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Chapter 5: Problem 56 Calculus: Graphical, Numerical, Algebraic 3
A town wants to drain and fill the small polluted swamp shown below. The swamp averages 5 ft deep. About how many cubic yards of dirt will it take to fill the area after the swamp is drained?
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Chapter 5: Problem 57 Calculus: Graphical, Numerical, Algebraic 3
We model the voltage V in our homes with the sine function V $ Vmax sin !120 !t", which expresses V in volts as a function of time t in seconds. The function runs through 60 cycles each second. The number Vmax is the peak voltage. To measure the voltage effectively, we use an instrument that measures the square root of the average value of the square of the voltage over a 1-second interval: Vrms $ $!%V2 %"av%. The subscript rms stands for root mean square. It turns out that Vrms $ % V $ ma 2% %x . (1) The familiar phrase 115 volts ac means that the rms voltage is 115. The peak voltage, obtained from Equation 1 as Vmax $ 115$%2, is about 163 volts. (a) Find the average value of V2 over a 1-sec interval. Then find Vrms, and verify Equation 1. (b) The circuit that runs your electric stove is rated 240 volts rms. What is the peak value of the allowable voltage?
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Chapter 5: Problem 58 Calculus: Graphical, Numerical, Algebraic 3
The rate at which water flows out of a pipe is given by a differentiable function R of time t. The table below records the rate at 4-hour intervals for a 24-hour period. (a) Use the Trapezoid Rule with 6 subdivisions of equal length to approximate !24 0 R(t)dt. Explain the meaning of your answer in terms of water flow, using correct units. (b) Is there some time t between 0 and 24 such that R!(t) " 0? Justify your answer. (c) Suppose the rate of water flow is approximated by Q(t) " 0.01(950 # 25x $ x2). Use Q(t) to approximate the average rate of water flow during the 24-hour period. Indicate units of measure
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Chapter 5: Problem 59 Calculus: Graphical, Numerical, Algebraic 3
Let f be a differentiable function with the following properties. i. f!(x) " ax2 # bx ii. f!(1) " $6 and f %!x" " 6 iii. !2 1 f (x) dx " 14 Find f (x). Show your work
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Chapter 5: Problem 60 Calculus: Graphical, Numerical, Algebraic 3
The graph of the function f, consisting of three line segments, is shown below. Let g(x) " !x 1 f (t)dt. (a) Compute g(4) and g($2). (b) Find the instantaneous rate of change of g, with respect to x, at x " 2. (c) Find the absolute minimum value of g on the closed interval [$2, 4]. Justify your answer. (d) The second derivative of g is not defined at x " 1 and x " 2. Which of these values are x-coordinates of points of inflection of the graph of g? Justify your answer
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