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Problem 5.2.37
Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function Ζ on [a,b]. Identify Ζ and express the limit as a definite integral.
n
lim β π*β (ln π*β) βπβ on [1,2]
β β 0 k=1
Problem 5.2.67
Use geometry and properties of integrals to evaluate
β«βΒΉ (2π + β(1βπΒ²) + 1) dπ
Problem 5.3.2
Suppose F is an antiderivative of Ζ and A is an area function of Ζ. What is the relationship between F and A?
Problem 5.2.65
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«ββ° Ζ(π) dπ
Problem 5.3.114
Max/min of area functions Suppose Ζ is continuous on [0 ,β) and A(π) is the net area of the region bounded by the graph of Ζ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of Ζ. Verify this fact with the function Ζ(π) = πΒ² - 10π.
Problem 5.3.55
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«Ο/β^Β³Ο/β΄ (cotΒ² π + 1) dπ
Problem 5.3.5
The linear function Ζ(π) = 3 β π is decreasing on the interval [0, 3]. Is its area function for Ζ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.
Problem 5.5.79
Variations on the substitution method Evaluate the following integrals.
β« π/(βπβ4) dπ
Problem 5.1.67
Identifying Riemann sums Fill in the blanks with an interval and a value of n.β
4
β Ζ (1.5 + k) β’ 1 is a midpoint Riemann sum for f on the interval [ ___ , ___ ]
k = 1
with n = ________ .
Problem 5.5.10
Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.
β« (6π + 1) β(3πΒ² + π) dπ , u = 3πΒ² + π
Problem 5.2.59
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βα΅ Ζ(π) dπ
Problem 5.5.18
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« πeΛ£Β² dπ
Problem 5.3.66
Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.
The region bounded by y = 6 cos π and the π-axis between π = βΟ/2 and π = Ο
Problem 5.2.79
Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.
β«βΒ² (2π + 1) dπ
Problem 5.1.19
Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
v = [1 / (2t + 1)] (m/s), for 0 β€ t β€ 8 ; n = 4
Problem 5.1.75
Displacement from velocity The following functions describe the velocity of a car (in mi/hr) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval [0,t], where 0 β€ t β€ 3.
v(t) = { 30 if 0 β€ t β€ 2
50 if 2 < t < 2.5
44 if 2.5 < t β€ 3
Problem 5.5.46
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«βΒΉ 2eΒ²Λ£ dπ
Problem 5.3.23
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure.
β«βΒΉ (πΒ² β 2π + 3) dπ
Problem 5.3.103
{Use of Tech} Areas of regions Find the area of the region π bounded by the graph of Ζ and the π-axis on the given interval. Graph Ζ and show the region π .
Ζ(π) = 2 β |π| on [ β 2 , 4]
Problem 5.1.63
{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.β
The midpoint Riemann sum for f(x) = xΒ³ on [3,11] with n = 32.
Problem 5.3.8
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Problem 5.5.59
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«β/ββ βββ^Β²/β΅ dπ/ xβ(25πΒ²β 1)
Problem 5.5.92
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β« π cosΒ²πΒ² dπ
Problem 5.2.83
Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.
β«ββ΄ (πΒ²β1) dπ
Problem 5.5.96
Areas of regions Find the area of the following regions.
The region bounded by the graph of Ζ(π) = x /β(πΒ² β9) and the π-axis between and π = 4 and π= 5
Problem 5.3.27
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.
β«ββ΅ (πΒ²β9) dπ
Problem 5.2.43
Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.
β«ββ΄ β(16β πΒ² ) dπ
Problem 5.4.47
Gateway Arch The Gateway Arch in St. Louis is 630 ft high and has a 630-ft base. Its shape can be modeled by the parabola y = 630 (1β (π/315)Β²) . Find the average height of the arch above the ground.
Problem 5.5.84
Variations on the substitution method Evaluate the following integrals.
β« (π΅ + 1) β(3π΅ + 2) dπ΅
Problem 5.5.49
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«βΒ³ ( 2Λ£ / 2Λ£ + 4 ) dπ
