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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 5.3.5
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 5.3.55
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«Ο/β^Β³Ο/β΄ (cotΒ² π + 1) dπ
Problem 5.1.65
Identifying Riemann sums Fill in the blanks with an interval and a value of n.β
4
β Ζ (1 + k) β’ 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with
k = 1
n = ________ .
Problem 5.4.1
If Ζ is an odd function, why is β«α΅ββ Ζ(π) dπ = 0?
Problem 5.1.11
Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length βπ? List the grid points xβ , xβ , xβ , xβ and xβ. Which points are used for the left, right, and midpoint Riemann sums?
Problem 5.1.17
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 = 2t + 1(m/s), for 0 β€ t β€ 8 ; n = 2
Problem 5.3.47
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«βΒ² 3/t dt
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.R.13
Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer.β
β«ββ΄ (πΒ³βπ) dπ
Problem 5.R.35
Find the intervals on which Ζ(π) = β«βΒΉ (tβ3) (tβ6)ΒΉΒΉ dt is increasing and the intervals on which it is decreasing.
Problem 5.R.113c
Function defined by an integral Let Ζ(π) = β«βΛ£ (t β 1)ΒΉβ΅ (tβ2)βΉ dt .
(c) For what values of π does Ζ have local minima? Local maxima?
Problem 5.R.57
Evaluating integrals Evaluate the following integrals.
β«βΒ² (2π + 1)Β³ dπ
Problem 5.R.66
Evaluating integrals Evaluate the following integrals.
β« π sin πΒ² cosβΈ πΒ² dπ
Problem 5.R.87
Area of regions Compute the area of the region bounded by the graph of Ζ and the π-axis on the given interval. You may find it useful to sketch the region.
Ζ(π)β = 16βπΒ² on [β4, 4]
Problem 5.R.89
Area of regions Compute the area of the region bounded by the graph of Ζ and the π-axis on the given interval. You may find it useful to sketch the region.
Ζ(π)β = 2 sin π/4 on [0, 2Ο]
Problem 5.R.1c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ and Ζ' are continuous functions for all real numbers.
(c) β«βα΅ Ζ'(π) dπ = Ζ(b) βΖ(a) .
Problem 5.R.109
Geometry of integrals Without evaluating the integrals, explain why the following statement is true for positive integers n:
β«βΒΉ πβΏdπ + β«βΒΉ βΏβ(πdπ) = 1
Problem 5.R.107
Limits with integrals Evaluate the following limits.
lim β«βΛ£ eα΅Β² dt
πβ2 ---------------
π β 2
Problem 5.R.91
Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of Ζ and the π-axis on the given interval. You may find it useful to sketch the region.β
Ζ(π) = πβ΄ β πΒ² on [β1, 1]
Problem 5.R.21
Properties of integrals Suppose β«ββ΄ Ζ(π) dπ = 6 , β«ββ΄ g(π) dπ = 4 and β«ββ΄ Ζ(π) dπ = 2 . Evaluate the following integrals or state that there is not enough information.
β«βΒ³ Ζ(π)/g(π) dπ
Problem 5.R.51
Evaluating integrals Evaluate the following integrals.
β« πΒ² cos πΒ³ dπ
Problem 5.R.86
Evaluating integrals Evaluate the following integrals.
β«ββ΅ |2πβ8|dπ
Problem 5.R.102e
Function defined by an integral Let H (π) = β«βΛ£ β(4 β tΒ²) dt, for β 2 β€ π β€ 2.
(e) Find the value of s such that H (π) = sH(βπ)
Problem 5.R.9c
Integration by Riemann sums Consider the integral β«ββ΄ (3πβ 2) dπ.
(c) Evaluate the definite integral by taking the limit as n ββ of the Riemann sum in part (b).
Problem 5.R.105f
Consider the function
Ζ(t) = { t if β2 β€ t < 0
tΒ²/2 if 0 β€ t β€ 2
and its graph shown below. Let F(π) = β«ββΛ£ Ζ(t) dt and G(π) = β«ββΛ£ Ζ(t) dt.
(f) Find a constant C such that F(π) = G(π) + C .
Problem 5.R.9b
Integration by Riemann sums Consider the integral β«ββ΄ (3πβ 2) dπ.
(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .
Problem 5.R.62
Evaluating integrals Evaluate the following integrals.
β« yΒ² /(yΒ³ + 27) dy
Problem 5.R.1g
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ and Ζ' are continuous functions for all real numbers.
(g) β« Ζ' (g(π))g' (π) d(π) = Ζ(g(π)) + C .
Problem 5.R.1d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ and Ζ' are continuous functions for all real numbers.
(d) If Ζ is continuous on [a,b] and β«βα΅ |Ζ(π)| dπ = 0 , then Ζ(π) = 0 on [a,b] .
