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Math Calculus CALCULUS -W/ACCESS

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. lim n → ∞ ∑ i = 1 n ( i 4 n 5 + i n 2 )

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. lim n → ∞ ∑ i = 1 n ( i 4 n 5 + i n 2 )

Solution Summary: The author explains how to evaluate the limit by first recognizing the sum as a Riemann sum.

CALCULUS -W/ACCESS

CALCULUS -W/ACCESS

8th Edition

ISBN: 9781305779075

Author: Stewart

Publisher: CENGAGE L

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Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

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Textbook Question

Chapter 4.3, Problem 67E

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].

lim n → ∞ ∑ i = 1 n ( i 4 n 5 + i n 2 )

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Chapter 4.3, Problem 66E

Chapter 4.3, Problem 68E

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Chapter 4 Solutions

CALCULUS -W/ACCESS

Ch. 4.1 - Prob. 1E Ch. 4.1 - a Use six rectangles to find estimates of each...Ch. 4.1 - a Estimate the area under the graph of f(x)=1/x...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Evaluate the upper and lower sums for...Ch. 4.1 - With a programmable calculator or a computer, it...Ch. 4.1 - Prob. 10E

Ch. 4.1 - Some computer algebra systems have commands that...Ch. 4.1 - Prob. 12E Ch. 4.1 - The speed of a runner increased steadily during...Ch. 4.1 - The table shows speedometer readings at 10-second...Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - The velocity graph of a braking car is shown. Use...Ch. 4.1 - Prob. 18E Ch. 4.1 - In someone infected with measles, the virus level...Ch. 4.1 - The table shows the number of people per day who...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Prob. 24E Ch. 4.1 - Determine a region whose area is equal to the...Ch. 4.1 - a Use Definition 2 to find an expression for the...Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.2 - Evaluate the Riemann sum for f(x)=x1,6x4, with...Ch. 4.2 - If f(x)=cosx0x3/4 evaluate the Riemann sum with n...Ch. 4.2 - If f(x)=x24,0x3, find the Riemann sum with n = 6,...Ch. 4.2 - a Find the Riemann sum for f(x)=1/x,1x2, with four...Ch. 4.2 - The graph of a function f is given. Estimate...Ch. 4.2 - The graph of g is shown. Estimate 24g(x)dx with...Ch. 4.2 - Prob. 7E Ch. 4.2 - The table gives the values of a function obtained...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - If you have a CAS that evaluates midpoint...Ch. 4.2 - With a programmable calculator or computer see the...Ch. 4.2 - Prob. 15E Ch. 4.2 - Prob. 16E Ch. 4.2 - Prob. 17E Ch. 4.2 - Prob. 18E Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Prob. 21E Ch. 4.2 - Prob. 22E Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Prob. 24E Ch. 4.2 - Prob. 25E Ch. 4.2 - Prob. 26E Ch. 4.2 - Prove that abxdx=b2a22 Ch. 4.2 - Prove that abx2dx=b3a33 Ch. 4.2 - Prob. 29E Ch. 4.2 - Express the integral as a limit of Riemann sums....Ch. 4.2 - Prob. 31E Ch. 4.2 - Express the integral as a limit of sums. Then...Ch. 4.2 - The graph of f is shown. Evaluate each integral by...Ch. 4.2 - The graph of g consists of two straight fines and...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Prob. 37E Ch. 4.2 - Prob. 38E Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Prob. 40E Ch. 4.2 - Prob. 41E Ch. 4.2 - Given that 0sin4xdx=38, what is 0sin4d?Ch. 4.2 - Prob. 43E Ch. 4.2 - Prob. 44E Ch. 4.2 - Prob. 45E Ch. 4.2 - Prob. 46E Ch. 4.2 - Prob. 47E Ch. 4.2 - If 28f(x)dx=7.3 and 24f(x)dx=5.9, find 48f(x)dx.Ch. 4.2 - If 09f(x)dx=37 and 09g(x)dx=16, find...Ch. 4.2 - Find 05f(x)dx if f(x)={3forx3xforx3 Ch. 4.2 - For the function / whose graph is shown, list the...Ch. 4.2 - If F(x)=2xf(t)dt, where f is the function whose...Ch. 4.2 - Each of the regions A, B, and C bounded by the...Ch. 4.2 - Suppose / has absolute minimum value m and...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Prob. 57E Ch. 4.2 - Prob. 58E Ch. 4.2 - Prob. 59E Ch. 4.2 - Prob. 60E Ch. 4.2 - Prob. 61E Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Prob. 63E Ch. 4.2 - Prob. 64E Ch. 4.2 - Use properties of integrals, together with...Ch. 4.2 - Prob. 66E Ch. 4.2 - Prob. 67E Ch. 4.2 - Prob. 68E Ch. 4.2 - Prob. 69E Ch. 4.2 - Prob. 70E Ch. 4.2 - Let f(x)=0 if x is any rational number and f(x)=1...Ch. 4.2 - Prob. 72E Ch. 4.2 - Express the limit as a definite intergal....Ch. 4.2 - Prob. 74E Ch. 4.2 - Find 12x2dx. Hint: Choose xi* to be the geometric...Ch. 4.3 - Explain exactly what is meant by the statement...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt where f is the function whose...Ch. 4.3 - Sketch the area represented by g(x). Then find...Ch. 4.3 - Sketch the area represented by g(x). Then find...Ch. 4.3 - Prob. 7E Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 16E Ch. 4.3 - Prob. 17E Ch. 4.3 - Prob. 18E Ch. 4.3 - Prob. 19E Ch. 4.3 - Evaluate the integral. 11x100dx Ch. 4.3 - Prob. 21E Ch. 4.3 - Prob. 22E Ch. 4.3 - Prob. 23E Ch. 4.3 - Prob. 24E Ch. 4.3 - Evaluate the integral. /6sind Ch. 4.3 - Evaluate the integral. 55dx Ch. 4.3 - Prob. 27E Ch. 4.3 - Evaluate the integral. 04(4t)tdt Ch. 4.3 - Evaluate the integral. 142+x2xdx Ch. 4.3 - Evaluate the integral. 12(3u2)(u+1)du Ch. 4.3 - Prob. 31E Ch. 4.3 - Evaluate the integral. /4/3csc2d Ch. 4.3 - Prob. 33E Ch. 4.3 - Evaluate the integral. 12s2+1s2ds Ch. 4.3 - Evaluate the integral. 12v5+3v6v4dv Ch. 4.3 - Prob. 36E Ch. 4.3 - Prob. 37E Ch. 4.3 - Prob. 38E Ch. 4.3 - Prob. 39E Ch. 4.3 - Prob. 40E Ch. 4.3 - Prob. 41E Ch. 4.3 - Prob. 42E Ch. 4.3 - Prob. 43E Ch. 4.3 - Prob. 44E Ch. 4.3 - Prob. 45E Ch. 4.3 - Prob. 46E Ch. 4.3 - Evaluate the integral and interpret it as a...Ch. 4.3 - Prob. 48E Ch. 4.3 - What is wrong with the equation? 21x4dx=x33]21=38 Ch. 4.3 - What is wrong with the equation? 124x3dx=2x2]12=32 Ch. 4.3 - Prob. 51E Ch. 4.3 - What is wrong with the equation? 0sec2xdx=tanx]0=0 Ch. 4.3 - Find the derivative of the function....Ch. 4.3 - Prob. 54E Ch. 4.3 - Prob. 55E Ch. 4.3 - Prob. 56E Ch. 4.3 - Let F(x)=xcosttdt. Find an equation of the tangent...Ch. 4.3 - Prob. 58E Ch. 4.3 - On what interval is the curve y=0xt2t2+t+2dt...Ch. 4.3 - Let F(x)=1xf(t)dt, where f is the function whose...Ch. 4.3 - Prob. 61E Ch. 4.3 - Prob. 62E Ch. 4.3 - The Fresnel function S was defined in Example 3...Ch. 4.3 - The sine integral function Si(x)=0xsinttdt is...Ch. 4.3 - Let g(x)=0xf(t)dt where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt where f is the function whose...Ch. 4.3 - Evaluate the limit by first recognizing the sum as...Ch. 4.3 - Evaluate the limit by first recognizing the sum as...Ch. 4.3 - Prob. 69E Ch. 4.3 - Prob. 70E Ch. 4.3 - a Show that 11+x31+x3forx0 b Show that...Ch. 4.3 - Prob. 72E Ch. 4.3 - Show that 0510x2x4+x2+1dx0.1 by comparing the...Ch. 4.3 - Let f(x)={0ifx0xif0x12xif1x20ifx2 and...Ch. 4.3 - Prob. 75E Ch. 4.3 - Prob. 76E Ch. 4.3 - A manufacturing company owns a major piece of...Ch. 4.3 - Prob. 78E Ch. 4.3 - The following exercises are intended only for...Ch. 4.3 - The following exercises are intended only for...Ch. 4.3 - Prob. 81E Ch. 4.3 - Prob. 82E Ch. 4.3 - Prob. 83E Ch. 4.3 - Prob. 84E Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Prob. 5E Ch. 4.4 - Prob. 6E Ch. 4.4 - Find the general indefinite integral....Ch. 4.4 - Prob. 8E Ch. 4.4 - Find the general indefinite integral....Ch. 4.4 - Prob. 10E Ch. 4.4 - Prob. 11E Ch. 4.4 - Prob. 12E Ch. 4.4 - Prob. 13E Ch. 4.4 - Prob. 14E Ch. 4.4 - Find the general indefinite integral....Ch. 4.4 - Prob. 16E Ch. 4.4 - Prob. 17E Ch. 4.4 - Prob. 18E Ch. 4.4 - Prob. 19E Ch. 4.4 - Prob. 20E Ch. 4.4 - Prob. 21E Ch. 4.4 - Prob. 22E Ch. 4.4 - Prob. 23E Ch. 4.4 - Evaluate the integral. 11t(1t)2dt Ch. 4.4 - Prob. 25E Ch. 4.4 - Prob. 26E Ch. 4.4 - Prob. 27E Ch. 4.4 - Prob. 28E Ch. 4.4 - Prob. 29E Ch. 4.4 - Prob. 30E Ch. 4.4 - Prob. 31E Ch. 4.4 - Prob. 32E Ch. 4.4 - Prob. 33E Ch. 4.4 - Prob. 34E Ch. 4.4 - Prob. 35E Ch. 4.4 - Prob. 36E Ch. 4.4 - Prob. 37E Ch. 4.4 - Prob. 38E Ch. 4.4 - Evaluate the integral. 25|x3|dx Ch. 4.4 - Prob. 40E Ch. 4.4 - Prob. 41E Ch. 4.4 - Prob. 42E Ch. 4.4 - Use a graph to estimate the x-intercepts of the...Ch. 4.4 - Prob. 44E Ch. 4.4 - The area of the region that lies to the right of...Ch. 4.4 - The boundaries of the shaded region in the figure...Ch. 4.4 - Prob. 47E Ch. 4.4 - Prob. 48E Ch. 4.4 - Prob. 49E Ch. 4.4 - A honeybee population starts with 100 bees and...Ch. 4.4 - Prob. 51E Ch. 4.4 - Prob. 52E Ch. 4.4 - Prob. 53E Ch. 4.4 - Prob. 54E Ch. 4.4 - The velocity function in meters per second is...Ch. 4.4 - Prob. 56E Ch. 4.4 - The acceleration function in m/s2 and the initial...Ch. 4.4 - Prob. 58E Ch. 4.4 - Prob. 59E Ch. 4.4 - Prob. 60E Ch. 4.4 - Prob. 61E Ch. 4.4 - Suppose that a volcano is erupting and readings of...Ch. 4.4 - Lake Lanier in Georgia, USA, is a reservoir...Ch. 4.4 - Prob. 64E Ch. 4.4 - The graph of the acceleration a(t) of a car...Ch. 4.4 - Shown is the graph of traffic on an Internet...Ch. 4.4 - Prob. 67E Ch. 4.4 - Prob. 68E Ch. 4.4 - Prob. 69E Ch. 4.4 - Prob. 70E Ch. 4.4 - Prob. 71E Ch. 4.4 - Prob. 72E Ch. 4.4 - Prob. 73E Ch. 4.4 - The area labeled B is three times the area labeled...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 3E Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 5E Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 7E Ch. 4.5 - Evaluate the indefinite integral. x2sin(x3)dx Ch. 4.5 - Prob. 9E Ch. 4.5 - Evaluate the indefinite integral. sin1+costdt Ch. 4.5 - Evaluate the indefinite integral. sin(2/3)d Ch. 4.5 - Evaluate the indefinite integral. sec22d Ch. 4.5 - Prob. 13E Ch. 4.5 - Evaluate the indefinite integral. y2(4y3)2/3dy Ch. 4.5 - Prob. 15E Ch. 4.5 - Prob. 16E Ch. 4.5 - Prob. 17E Ch. 4.5 - Prob. 18E Ch. 4.5 - Prob. 19E Ch. 4.5 - Prob. 20E Ch. 4.5 - Prob. 21E Ch. 4.5 - Prob. 22E Ch. 4.5 - Prob. 23E Ch. 4.5 - Prob. 24E Ch. 4.5 - Prob. 25E Ch. 4.5 - Evaluate the indefinite integral. sec2xtan2xdx Ch. 4.5 - Evaluate the indefinite integral. sec3xtanxdx Ch. 4.5 - Prob. 28E Ch. 4.5 - Evaluate the indefinite integral. x(2x+5)8dx Ch. 4.5 - Prob. 30E Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Prob. 32E Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Prob. 35E Ch. 4.5 - Prob. 36E Ch. 4.5 - Evaluate the definite integral. 011+7x3dx Ch. 4.5 - Prob. 38E Ch. 4.5 - Evaluate the integral. 0/6sintcos2tdt Ch. 4.5 - Prob. 40E Ch. 4.5 - Evaluate the definite integral. /4/4(x3+x4tanx)dx Ch. 4.5 - Prob. 42E Ch. 4.5 - Prob. 43E Ch. 4.5 - Prob. 44E Ch. 4.5 - Prob. 45E Ch. 4.5 - Evaluate the definite integral. /3/3x4sinxdx Ch. 4.5 - Evaluate the definite integral. 12xx1dx Ch. 4.5 - Prob. 48E Ch. 4.5 - Evaluate the definite integral. 1/21cos(x2)x3dx Ch. 4.5 - Prob. 50E Ch. 4.5 - Evaluate the definite integral. 01dx(1+x)4 Ch. 4.5 - Prob. 52E Ch. 4.5 - Prob. 53E Ch. 4.5 - Prob. 54E Ch. 4.5 - Prob. 55E Ch. 4.5 - Prob. 56E Ch. 4.5 - Prob. 57E Ch. 4.5 - Prob. 58E Ch. 4.5 - Prob. 59E Ch. 4.5 - Prob. 60E Ch. 4.5 - Prob. 61E Ch. 4.5 - If f is continuous function on , prove that...Ch. 4.5 - If a and b are positive numbers, show that...Ch. 4.5 - Prob. 64E Ch. 4.5 - Prob. 65E Ch. 4.5 - Prob. 66E Ch. 4.5 - Prob. 67E Ch. 4.5 - Prob. 68E Ch. 4.5 - Prob. 69E Ch. 4.5 - Prob. 70E Ch. 4.5 - Prob. 71E Ch. 4.5 - Prob. 72E Ch. 4.5 - Prob. 73E Ch. 4.5 - Prob. 74E Ch. 4.5 - Prob. 75E Ch. 4.5 - Prob. 76E Ch. 4.5 - Prob. 77E Ch. 4.5 - Prob. 78E Ch. 4.5 - Prob. 79E Ch. 4.5 - Prob. 80E Ch. 4.5 - Prob. 81E Ch. 4.5 - Prob. 82E Ch. 4.5 - Prob. 83E Ch. 4.5 - Prob. 84E Ch. 4.5 - Prob. 85E Ch. 4.R - a Write an expression for a Riemann sum of a...Ch. 4.R - a Write the definition of the definite integral of...Ch. 4.R - Prob. 3CC Ch. 4.R - Prob. 4CC Ch. 4.R - Prob. 5CC Ch. 4.R - Prob. 6CC Ch. 4.R - a Explain the meaning of the indefinite integral...Ch. 4.R - Explain exactly what is meant by the statement...Ch. 4.R - State the Substitution Rule. 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