Calculus & Its Applications (14th Edition)
Calculus & Its Applications (14th Edition)
14th Edition
ISBN: 9780134437774
Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar
Publisher: PEARSON
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Riemann Sum
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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Chapter 6.3, Problem 1CYU

Repeat Example 6 using midpoints of the subintervals.

Example 6

Improving a Riemann Sum Approximation

Use a Riemann sum with n = 10 to estimate the area under the graph of f ( x ) = x 2 from 0 to 1 . Select the right endpoints of the subintervals as x 1 ,     x 2 ,     … ,     x 10 .

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Chapter 6 Solutions

Calculus & Its Applications (14th Edition)

Ch. 6.1 - Determine the following: a. t7/2dt b....Ch. 6.1 - Find a function f(t) that satisfies f(t)=3t+5 and...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Find all antiderivatives of each following...Ch. 6.1 - Determine the following: 4x3dxCh. 6.1 - Determine the following: 13xdx
Ch. 6.1 - Determine the following: 7dxCh. 6.1 - Determine the following: k2dx ((kisaconstant).Ch. 6.1 - Determine the following: xcdx(cisaconstant0)...Ch. 6.1 - Determine the following: xx2dx.Ch. 6.1 - Determine the following: (2x+x2)dx.Ch. 6.1 - Determine the following: 17xdx.Ch. 6.1 - Determine the following: xxdx.Ch. 6.1 - Determine the following: (2x+2x)dx.Ch. 6.1 - Determine the following: (x2x2+13x)dx.Ch. 6.1 - Determine the following: (72x3x3)dx.Ch. 6.1 - Determine the following: 3e2xdx.Ch. 6.1 - Determine the following: exdx.Ch. 6.1 - Determine the following: edx.Ch. 6.1 - Determine the following: 72e2xdx.Ch. 6.1 - Determine the following: 2(e2x+1)dx.Ch. 6.1 - Determine the following: (3ex+2xe0.5x2)dx.Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - In Exercises 25-36, find the value of k that makes...Ch. 6.1 - Find all functions f(t) that satisfy the given...Ch. 6.1 - Find all functions f(t) that satisfy the given...Ch. 6.1 - Find all functions f(t) that satisfy the given...Ch. 6.1 - Find all functions f(t) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Find all functions f(x) that satisfy the given...Ch. 6.1 - Figure 4 shows the graphs of several functions...Ch. 6.1 - Figure 5 shows the graphs of several functions...Ch. 6.1 - Which of the following is lnxdx ? a.1x+C b.xlnxx+C...Ch. 6.1 - Which of the following is xx+1dx?...Ch. 6.1 - Figure 6 contains the graph of a function F(x). On...Ch. 6.1 - Figure 7 contains an antiderivative of the...Ch. 6.1 - The function g(x) in Fig. 8, resulted from...Ch. 6.1 - The function g(x) in Fig.9 resulted from shifting...Ch. 6.1 - Height of a Ball A ball is thrown upward from a...Ch. 6.1 - Free Fall A rock is dropped from the top of a...Ch. 6.1 - Rate of Production Let P(t) be the total output of...Ch. 6.1 - Rate of Production After t hours of operation, a...Ch. 6.1 - Heat DiffusionA package of frozen strawberries is...Ch. 6.1 - Epidemic A flu epidemic hits a town. Let P(t) be...Ch. 6.1 - Profit A small tie shop finds that at a sales...Ch. 6.1 - Prob. 62ECh. 6.1 - Prob. 63ECh. 6.1 - U.S. Natural Gas Production Since 1987, the rate...Ch. 6.1 - Prob. 65ECh. 6.1 - Prob. 66ECh. 6.1 - Prob. 67ECh. 6.1 - Prob. 68ECh. 6.2 - Evaluate 01e2x1exdx.Ch. 6.2 - If f(t)=1t, find f(2)f(0).Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - In Exercises 114, evaluate the given integral....Ch. 6.2 - Given 01f(x)dx=3.5 and 14f(x)dx=5, find 04f(x)dx.Ch. 6.2 - Given 11f(x)dx=0 and 110f(x)dx=4, find 110f(x)dx.Ch. 6.2 - Given 13f(x)dx=3 and 13g(x)dx=1, find...Ch. 6.2 - Given 0.53f(x)dx=0 and 0.53(2g(x)+f(x))dx=4, find...Ch. 6.2 - In Exercises 1922, combine the integrals into one...Ch. 6.2 - In Exercises 1922, combine the integrals into one...Ch. 6.2 - In Exercises 1922, combine the integrals into one...Ch. 6.2 - In Exercises 1922, combine the integrals into one...Ch. 6.2 - In Exercises 2326, use formula (8) to help you...Ch. 6.2 - In Exercises 2326, use formula (8) to help you...Ch. 6.2 - In Exercises 2326, use formula (8) to help you...Ch. 6.2 - In Exercises 2326, use formula (8) to help you...Ch. 6.2 - Refer to Fig. 4 and evaluate 02f(x)dx. Figure 4Ch. 6.2 - Refer to Fig. 5 and evaluate 03f(x)dx. Figure 5Ch. 6.2 - Refer to Fig. 6 and evaluate 11f(t)dt. Figure 6Ch. 6.2 - Refer to Fig. 7 and evaluate 12f(t)dt. Figure 7Ch. 6.2 - Net Change in Position A rock is dropped from the...Ch. 6.2 - Net change in Position The velocity at time t...Ch. 6.2 - Net Change in Position The velocity at time t...Ch. 6.2 - Velocity of a Skydiver The velocity of a skydiver...Ch. 6.2 - Net Change in Cost A companys marginal cost...Ch. 6.2 - Prob. 36ECh. 6.2 - Net Increase of an Investment An investment grew...Ch. 6.2 - Depreciation of Real Estate A property with an...Ch. 6.2 - Population Model with Emigration The rate of...Ch. 6.2 - Paying Down a Mortgage You took a 200,000 home...Ch. 6.2 - Mortgage Using the data from the previous...Ch. 6.2 - Radioactive Decay A sample of radioactive material...Ch. 6.2 - Prob. 43ECh. 6.2 - Level of Water in a Tank A conical-shaped tank is...Ch. 6.3 - Repeat Example 6 using midpoints of the...Ch. 6.3 - Repeat Example 6 using left endpoints of the...Ch. 6.3 - In exercises 16, compute the area of the shaded...Ch. 6.3 - In exercises 16, compute the area of the shaded...Ch. 6.3 - In exercise 16, compute the area of the shaded...Ch. 6.3 - In exercise 16, compute the area of the shaded...Ch. 6.3 - In exercise 16, compute the area of the shaded...Ch. 6.3 - In exercise 16, compute the area of the shaded...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - In exercises 712, set-up the definite integral...Ch. 6.3 - Prob. 12ECh. 6.3 - Prob. 13ECh. 6.3 - Prob. 14ECh. 6.3 - Prob. 15ECh. 6.3 - In Exercises 1618, draw the region whose area is...Ch. 6.3 - In Exercises 1618, draw the region whose area is...Ch. 6.3 - In Exercises 1618, draw the region whose area is...Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Find the area under each of the given curves....Ch. 6.3 - Prob. 25ECh. 6.3 - Find the real number b0 so that the area under the...Ch. 6.3 - Determine x and the midpoints of the subintervals...Ch. 6.3 - Determine x and the midpoints of the subintervals...Ch. 6.3 - Determine x and the midpoints of the subintervals...Ch. 6.3 - Determine x and the midpoints of the subintervals...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3136, use a Riemann sum to...Ch. 6.3 - In Exercises 3740, use a Riemann sum to...Ch. 6.3 - In Exercises 3740, use a Riemann sum to...Ch. 6.3 - In Exercises 3740, use a Riemann sum to...Ch. 6.3 - Prob. 40ECh. 6.3 - Use a Riemann sum with n=4 and left endpoints to...Ch. 6.3 - Prob. 42ECh. 6.3 - The graph of the function f(x)=1x2 on the interval...Ch. 6.3 - Use a Riemann sum with n=5 and midpoints to...Ch. 6.3 - Estimate the area (in square feet) of the...Ch. 6.3 - Prob. 46ECh. 6.3 - Prob. 47ECh. 6.3 - Prob. 48ECh. 6.3 - Technology Exercises. The area under the graph of...Ch. 6.3 - Prob. 50ECh. 6.3 - Prob. 51ECh. 6.3 - Prob. 52ECh. 6.4 - Find the area between the curves y=x+3 and...Ch. 6.4 - A company plans to increase its production from 10...Ch. 6.4 - Write a definite integral or sum of definite...Ch. 6.4 - Write a definite integral or sum of definite...Ch. 6.4 - Shade the portion of Fig. 23 whose area is given...Ch. 6.4 - Shade the portion ofFig. 24 whose area is given by...Ch. 6.4 - Let f(x) be the function pictured in Fig. 25....Ch. 6.4 - Let g(x) be the function pictured in Fig. 26....Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curve and...Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region between the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region bounded by the curves....Ch. 6.4 - Find the area of the region between y=x23x and the...Ch. 6.4 - Find the area of the region between y=x2 and...Ch. 6.4 - Find the area in Fig. 27 of the region bounded by...Ch. 6.4 - Find the area of the region bounded by y=1/x,y=4x...Ch. 6.4 - Height of a Helicopter A helicopter is rising...Ch. 6.4 - Assembly line productionAfter t hour of operation,...Ch. 6.4 - Cost Suppose that the marginal cost function for a...Ch. 6.4 - ProfitSuppose that the marginal profit function...Ch. 6.4 - Marginal Profit Let M(x) be a companys marginal...Ch. 6.4 - Marginal Profit Let M(x) be a companys marginal...Ch. 6.4 - Prob. 37ECh. 6.4 - VelocitySuppose that the velocity of a car at time...Ch. 6.4 - Deforestation and Fuel wood Deforestation is one...Ch. 6.4 - Refer to Exercise 39. The rate of new tree growth...Ch. 6.4 - After an advertising campaign, a companys marginal...Ch. 6.4 - Profit and Area The marginal profit for a certain...Ch. 6.4 - Velocity and Distance Two rockets are fired...Ch. 6.4 - Distance TraveledCars A and B start at the same...Ch. 6.4 - Displacement versus Distance Traveled The velocity...Ch. 6.4 - Displacement versus Distance Traveled The velocity...Ch. 6.4 - Prob. 47ECh. 6.4 - Prob. 48ECh. 6.4 - Prob. 49ECh. 6.4 - Prob. 50ECh. 6.5 - A rock dropped from a bridge has a velocity of 32t...Ch. 6.5 - An Investment yields a continuous income stream of...Ch. 6.5 - Determine the average value of f(x) over the...Ch. 6.5 - Determine the average value of f(x) over the...Ch. 6.5 - Determine the average value of f(x) over the...Ch. 6.5 - Prob. 4ECh. 6.5 - Determine the average value of f(x) over the...Ch. 6.5 - Prob. 6ECh. 6.5 - Average Temperature During a certain 12-hour...Ch. 6.5 - Average PopulationAssuming that a countrys...Ch. 6.5 - Average Amount of Radium. One hundred grams of...Ch. 6.5 - Average Amount of Money. One hundred dollars is...Ch. 6.5 - Consumers Surplus Find the consumers surplus for...Ch. 6.5 - Consumers Surplus Find the consumers surplus for...Ch. 6.5 - Consumers Surplus Find the consumers surplus for...Ch. 6.5 - Consumers Surplus Find the consumers surplus for...Ch. 6.5 - Prob. 15ECh. 6.5 - Prob. 16ECh. 6.5 - Prob. 17ECh. 6.5 - Prob. 18ECh. 6.5 - Prob. 19ECh. 6.5 - Prob. 20ECh. 6.5 - Prob. 21ECh. 6.5 - Prob. 22ECh. 6.5 - Prob. 23ECh. 6.5 - Future Value Suppose that money is deposited...Ch. 6.5 - Prob. 25ECh. 6.5 - Prob. 26ECh. 6.5 - Volume of Solids of Revolution Find the volume of...Ch. 6.5 - Volume of Solids of Revolution Find the volume of...Ch. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.5 - Prob. 31ECh. 6.5 - Prob. 32ECh. 6.5 - Prob. 33ECh. 6.5 - Prob. 34ECh. 6.5 - Prob. 35ECh. 6.5 - Prob. 36ECh. 6.5 - Prob. 37ECh. 6.5 - For the Riemann sum...Ch. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 6.5 - Prob. 41ECh. 6.5 - Prob. 42ECh. 6.5 - Prob. 43ECh. 6.5 - Prob. 44ECh. 6.5 - Prob. 45ECh. 6.5 - Prob. 46ECh. 6 - What does it mean to antidifferentiate a function?Ch. 6 - Prob. 2CCECh. 6 - Prob. 3CCECh. 6 - Prob. 4CCECh. 6 - Prob. 5CCECh. 6 - Prob. 6CCECh. 6 - Prob. 7CCECh. 6 - Prob. 8CCECh. 6 - Prob. 9CCECh. 6 - Prob. 10CCECh. 6 - Prob. 11CCECh. 6 - Calculate the following integrals. 32dxCh. 6 - Prob. 2RECh. 6 - Calculate the following integrals. x+1dxCh. 6 - Calculate the following integrals. 2x+4dxCh. 6 - Calculate the following integrals. 2(x3+3x21)dxCh. 6 - Calculate the following integrals. x+35dxCh. 6 - Calculate the following integrals. ex/2dxCh. 6 - Calculate the following integrals. 5x7dxCh. 6 - Calculate the following integrals. (3x44x3)dxCh. 6 - Calculate the following integrals. (2x+3)7dxCh. 6 - Calculate the following integrals. 4xdxCh. 6 - Calculate the following integrals. (5xx5)dxCh. 6 - Calculate the following integrals. 11(x+1)2dxCh. 6 - Calculate the following integrals. 01/8x3dxCh. 6 - Calculate the following integrals. 122x+4dxCh. 6 - Calculate the following integrals. 201(2x+11x+4)dxCh. 6 - Calculate the following integrals. 124x5dxCh. 6 - Calculate the following integrals. 2308x+1dxCh. 6 - Calculate the following integrals. 141x2dxCh. 6 - Calculate the following integrals. 36e2(x/3)dxCh. 6 - Calculate the following integrals. 05(5+3x)1dxCh. 6 - Calculate the following integrals. 2232e3xdxCh. 6 - Calculate the following integrals. 0ln2(exex)dxCh. 6 - Calculate the following integrals. ln2ln3(ex+ex)dxCh. 6 - Calculate the following integrals. 0ln3ex+exe2xdxCh. 6 - Calculate the following integrals. 013+e2xexdxCh. 6 - Find the area under the curve y=(3x2)3 from x=1 to...Ch. 6 - Find the area under the curve y=1+x from x=1 to...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - In Exercises 2936, Find the area of the shaded...Ch. 6 - Find the area of the region bounded by the curves...Ch. 6 - Find the area of the region between the curves...Ch. 6 - Find the function f(x) for which...Ch. 6 - Find the function f(x) for which f(x)=e5x,f(0)=1.Ch. 6 - Describe all solutions of the following...Ch. 6 - Let k be a constant, and let y=f(t) be a function...Ch. 6 - Prob. 43RECh. 6 - Prob. 44RECh. 6 - A drug is injected into a patient at the rate of...Ch. 6 - A rock thrown straight up into the air has a...Ch. 6 - Prob. 47RECh. 6 - Prob. 48RECh. 6 - Prob. 49RECh. 6 - Prob. 50RECh. 6 - Find the consumers surplus for the demand curve...Ch. 6 - Three thousand dollars is deposited in the bank at...Ch. 6 - Find the average value of f(x)=1/x3 from x=13 to...Ch. 6 - Prob. 54RECh. 6 - In Fig. 2, three regions are labelled with their...Ch. 6 - Prob. 56RECh. 6 - Prob. 57RECh. 6 - Prob. 58RECh. 6 - Prob. 59RECh. 6 - Prob. 60RECh. 6 - Prob. 61RECh. 6 - Prob. 62RECh. 6 - Prob. 63RECh. 6 - Prob. 64RECh. 6 - Prob. 65RECh. 6 - Prob. 66RECh. 6 - Prob. 67RECh. 6 - Prob. 68RECh. 6 - Prob. 69RECh. 6 - Prob. 70RECh. 6 - Prob. 71RECh. 6 - Prob. 72RECh. 6 - Prob. 73RECh. 6 - Prob. 74RE
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