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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

How much work is done raising a 4-kg mass to a height of 16 m above ground?
Compute the area of the region in Figure 1(B) enclosed by y = 2 − x2 and y = x. -2 y -2 (B) 1 y=x y=2-x² X
Is∫ba (ƒ(x) − g(x)) dx still equal to the area between the graphs of ƒ and g if ƒ(x) ≥ 0 but g(x) ≤ 0?
Find the area of the region between the graphs of ƒ(x) = 3x + 8 and g(x) = x2 + 2x + 2 over [0, 2] (Figure 13). 20- 10- f(x) = 3x+8 1 g(x) = x² + 2x+2 2 3 -X
Let V be the volume of a solid of revolution about the y-axis.(a) Does the Shell Method for computing V lead to an integral with respect to x or y?(b) Does the Disk or Washer Method for computing V lead to an integral with respect to x or y?
Sketch the solid obtained by rotating the region underneath the graph of the function over the given interval about the y-axis, and find its volume.ƒ(x) = √x, [0, 4]
Find the volume of a solid extending from y = 2 to y = 5 if every cross section has area A(y) = 5.
True or false? When the region under a single graph is rotated about the x-axis, the cross sections of the solid perpendicular to the x-axis are circular disks.
(a) Sketch the solid obtained by revolving the region under the graph of ƒ about the x-axis over the given interval, (b) Describe the cross section perpendicular to the x-axis located at x, and (c) Calculate the volume of the solid.ƒ(x) = x2, [1, 3]
Why is integration needed to compute the work performed in pumping water out of a tank but not to compute the work performed in lifting up the tank?
Use the method of Exercise 2 to find the formula for the volume of a right circular cone of height h whose base is a circle of radius R [Figure 16(B)].Data From Exercise 2Let V be the volume of a right circular cone of height 10 whose base is a circle of radius 4 [Figure 16(A)].(a) Use similar
How much work is done raising a 4-lb mass to a height of 16 ft above ground?
Suppose that ƒ(x) ≥ g(x) on [0, 3] and g(x) ≥ ƒ(x) on [3, 5]. Express the area between the graphs over [0, 5] as a sum of integrals.
Find the area of the region enclosed by the graphs of ƒ(x) = x2 + 2 and g(x) = 2x + 5 (Figure 14). y 10+ g(x)= 2x+5 f(x)=x² +2 2 +X 3
If we rotate the region under the curve y = 8 between x = 2 and x = 3 about the x-axis, what answer should the Shell Method give us?
Sketch the solid obtained by rotating the region underneath the graph of the function over the given interval about the y-axis, and find its volume.ƒ(x) = x−1, [1, 3]
Which of the following represents the work required to stretch a spring (with spring constant k) a distance x beyond its equilibrium position: kx, -kx, 1 mk², ½kx², or mx²?
What is the definition of flow rate?
(a) Sketch the solid obtained by revolving the region under the graph of ƒ about the x-axis over the given interval, (b) Describe the cross section perpendicular to the x-axis located at x, and (c) Calculate the volume of the solid.ƒ(x) = √x + 1, [1, 4]
Compute the work (in joules) required to stretch or compress a spring as indicated, assuming a spring constant of k = 800 N/m.Stretching from equilibrium to 12 cm past equilibrium
Calculate the volume of the ramp in Figure 17 in three ways by integrating the area of the cross sections:(a) Perpendicular to the x-axis (rectangles)(b) Perpendicular to the y-axis (triangles)(c) Perpendicular to the z-axis (rectangles) 2 4 6 -y X
True or false? When the region between two graphs is rotated about the x-axis, the cross sections of the solid perpendicular to the x-axis are circular disks.
Suppose that the graph of x = ƒ(y) lies to the left of the y-axis. Is ∫ba ƒ(y) dy positive or negative?
Find the area of the region enclosed by the graphs of ƒ(x) = x3 − 10x and g(x) = 6x (Figure 15). -4 + -2 20 y -20+ f(x) = x³ - 10x 2 4 g(x) = 6x
Find the area of the region enclosed by the graphs of the functions.y = x2 + 2x, y = x2 − 1, h(x) = x2 + x − 2
Sketch the solid obtained by rotating the region underneath the graph of the function over the given interval about the y-axis, and find its volume.ƒ(x) = 4 − x2, [0, 2]
Draw a graph of the signed area represented by the integral and compute it using geometry. 3 J-2 |x|dx
Draw a graph of the signed area represented by the integral and compute it using geometry. L₂2-1 J-2 (2-x) dx
Calculate in two ways:(a) As the limit (b) By using geometry Le -1 (4x - 8) dx
Sketch the signed area represented by the integral. Indicate the regions of positive and negative area. Зл sin x dx
Sketch the signed area represented by the integral. Indicate the regions of positive and negative area. So 0 (124x4) dx
Sketch the signed area represented by the integral. Indicate the regions of positive and negative area. S₂ (1² − 1)(1² – 4) dt -2
Determine the sign of the integral without calculating it. Draw a graph if necessary. J-2 x dx
Determine the sign of the integral without calculating it. Draw a graph if necessary. J-2 x³ dx
Determine the sign of the integral without calculating it. Draw a graph if necessary. -27 x sin x dx
Use properties of the integral and the formulas in the summary to calculate the integrals. So 0 (6t - 3) dt
Use properties of the integral and the formulas in the summary to calculate the integrals. L₁ (4x 1-3 (4x + 7) dx
Use properties of the integral and the formulas in the summary to calculate the integrals. So 10 x dx
Use properties of the integral and the formulas in the summary to calculate the integrals. J2 xdx
Use properties of the integral and the formulas in the summary to calculate the integrals. S' (u² - Jo (u² - 2u) du
Use properties of the integral and the formulas in the summary to calculate the integrals. 10 2 1/2 (12y² + 6y) dy
Use properties of the integral and the formulas in the summary to calculate the integrals. J-3 (7t² + t + 1) dt
Use properties of the integral and the formulas in the summary to calculate the integrals. S (9x - 4x²) dx -3
Use properties of the integral and the formulas in the summary to calculate the integrals. L'az -a 2 (x² + x) dx
Calculate the integral, assuming that So. f(x) dx = 5, S. 8(x g(x) dx = 12
Use properties of the integral and the formulas in the summary to calculate the integrals. -- a x² dx
Calculate the integral, assuming that So. f(x) dx = 5, S. 8(x g(x) dx = 12
Calculate the integral, assuming that So. f(x) dx = 5, S. ²8 (² g(x) dx = 12
Assume a Find an expression for ∫ba H(x) dx in terms of a and b. H(x) = 8 0 when x < 0 when x > 0 1
By computing the limit of right-endpoint approximations, prove that if b > 0, then 6² x³ dx = 54.
Using the result of Exercise 50, prove that for all b (negative, zero, and positive),Data From Exercise 50By computing the limit of right-endpoint approximations, prove that if b > 0, then So = xp x b4 4
Evaluate the integral using the formulas in the summary and Eq. (1).Equation (1) distance traveled velocity x time elapsed VAI
Evaluate the integral using the formulas in the summary and Eq. (1).Equation (1) distance traveled = velocity x time elapsed VAI
Evaluate the integral using the formulas in the summary and Eq. (1).Equation (1) distance traveled= velocity x time elapsed VAI
Evaluate the integral using the formulas in the summary and Eq. (1).Equation (1) distance traveled velocity x time elapsed VAI
Evaluate the integral using the formulas in the summary and Eq. (1).Equation (1) distance traveled = velocity x time elapsed VAI
Calculate the integral, assuming that S's f(x) dx = 1, S² f(x) dx = 4, Sfe f(x) dx = 7
Calculate the integral, assuming that S's f(x) dx = 1, S² f(x) dx = 4, Sfe f(x) dx = 7
Calculate the integral, assuming that S's f(x) dx = 1, S² f(x) dx = 4, Sfe f(x) dx = 7
Calculate the integral, assuming that S's f(x) dx = 1, S² f(x) dx = 4, Sfe f(x) dx = 7
Express each integral as a single integral. Jo f(x) dx + J3 f(x) dx
Express each integral as a single integral. J2 ·S₁₁ f(x) dx - f(x) dx
Express each integral as a single integral. S f(x) dx - x-S₁² ² 2 f(x) dx
Express each integral as a single integral. J7 f(x) dx + J3 f(x) dx
Prove the relationship for arbitrary a and b using the formulas in the summary. Ja xdx= 62 b²-a² 2
Prove the relationship for arbitrary a and b using the formulas in the summary. D, x² dx = 63 b³a³ 3
Explain the difference in graphical interpretation between S f(x) dx and So If(x)|dx.
Use the graphical interpretation of the definite integral to explain the inequalitywhere ƒ is continuous. Explain also why equality holds if and only if either ƒ(x) ≥ 0 for all x or ƒ(x) ≤ 0 for all x. [ 1500 S sa f(x) dx ≤ x) dx : |f(x)|dx
Let ƒ(x) = x. Find an interval [a, b] such that || | * 5(x) dx| = 1/2 and Sov = xp|(x).fl 3 2
Determine the sign of the integral without calculating it. Draw a graph if necessary. 10 27 sin x X dx
Repeat Exercise 7 for ƒ(x) = 20 − 3x over [2, 4].Data From Exercise 7Let ƒ(x) = 2x + 3.(a) Compute R6 and L6 over [0, 3].(b) Use geometry to find the exact area A and compute the errors |A − R6| and |A − L6| in the approximations.
Write the sum in summation notation. π sin(7) + sin (7) + sin (3) + ... + sin( n (² n+ 1
Calculate R3 and L3 for ƒ(x) = x2 − x + 4 over [1, 4]. Then sketch the graph of ƒ and the rectangles that make up each approximation. Is the area under the graph larger or smaller than R3? Is it larger or smaller than L3?
Calculate the sums: IMU (b) A (C)
Calculate the sums: (a) 4 ∑sin() j=3 (b) k=3 1 k-1 (c) j=0 3-1
Let b1 = 4, b2 = 1, b3 = 2, and b4 = −4. Calculate: (a) i=2 bi 2 (0) Σαβ - 6;) j=1 3 ο Σκόκ k=1
Calculate Expand and use formulas (3) and (4). 200 Σ j-101
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 20 Σ8;3 j=1
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 20 Σ8;3 j=1
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 30 k=1 (4k – 3)
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 150 Ση n n=51
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 200 ΣΕ k=101
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 50 j=0 jj-1)
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 30 j=2 j+ 4j²2 3
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 30 (4-m)³ m=1
Use linearity and formulas (3)–(5) to rewrite and evaluate the sums. 20 Σ m=1 5+ 3m 2 2
Use formulas (3)–(5) to evaluate the limit. N Σ i=1 i lim N-00 N2
Use formulas (3)–(5) to evaluate the limit. lim N→∞0 N j=1 N4
Use formulas (3)–(5) to evaluate the limit. lim N→∞0 N i=1 i² - i + 1 N³
Use formulas (3)–(5) to evaluate the limit. lim N→00 N i=1 1³ N4 20 N
Calculate the limit for the given function and interval. Verify your answer by using geometry. lim RN, f(x) = 9x, [0, 2] N→∞0
Calculate the limit for the given function and interval. Verify your answer by using geometry. lim RN, f(x) = 3x + 6, [1,4] N→∞0
Calculate the limit for the given function and interval. Verify your answer by using geometry. lim Ly, f(x)= x +2, [0,4] N→∞0⁰
Calculate the limit for the given function and interval. Verify your answer by using geometry. lim Ly, f(x) = 4x-2, [1,3] N→∞0
Calculate the limit for the given function and interval. Verify your answer by using geometry. lim MN, f(x) = x, [0, 2] N-00
Calculate the limit for the given function and interval. Verify your answer by using geometry. lim MN, f(x) = 12 - 4x, [2, 6] N→00
Show, for ƒ(x) = 3x2 + 4x over [0, 2], thatThen evaluate RN = 2 N 12j² 8j + N² N
Show, for ƒ(x) = 3x3 − x2 over [1, 5], thatThen evaluate RN # Σ (192/² + 128/² 28/+2) + N N3
Find a formula for RN and compute the area under the graph as a limit. f(x)=x², [1,3]
Find a formula for RN and compute the area under the graph as a limit. f(x) = x², [-1,5]

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