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Problem 5.R.96a
Velocity to displacement An object travels on the π-axis with a velocity given by v(t) = 2t + 5, for 0 β€ t β€ 4.
(a) How far does the object travel, for 0 β€ t β€ 4 ?
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] .
Problem 5.R.48
Evaluating integrals Evaluate the following integrals.
β«ββ΄ ((βv + v) / v ) dv
Problem 5.R.18
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.
ββ«βΒΉ 2Ζ(π) dπ
Problem 5.R.23a
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(a) β«ββ΄ Ζ(π) dπ
Problem 5.R.23c
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(c) β«β β· Ζ(π) dπ
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.104
Change of variables Use the change of variables uΒ³ = πΒ² β 1 to evaluate the integral β«βΒ³ πβ(πΒ²β1) dπ .
Problem 5.R.102a
Function defined by an integral Let H (π) = β«βΛ£ β(4 β tΒ²) dt, for β 2 β€ π β€ 2.
(a) Evaluate H (0) .
Problem 5.R.25
Use geometry and properties of integrals to evaluate the following definite integrals.β
β«ββ΄ β(8πβπΒ²) dπ . (Hint: Complete the square .)
Problem 5.R.60
Evaluating integrals Evaluate the following integrals.
β« sin π΅ sin (cos π΅) dπ΅
Problem 5.R.81
Evaluating integrals Evaluate the following integrals.
β«ββ/β ^Β²/β΅ dπ/πβ(25πΒ² β1)
Problem 5.R.99b
(b) Find the average value of Ζ shown in the figure on the interval [2,6] and then find the point(s) c in (2, 6) guaranteed to exist by the Mean Value Theorem for Integrals.
Problem 5.R.9d
Integration by Riemann sums Consider the integral β«ββ΄ (3πβ 2) dπ.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
Problem 5.R.71
Evaluating integrals Evaluate the following integrals.
β«ββ β΅ ΟΒ³ /β(Οβ΅β° + ΟΒ²β° + 1) dΟ (Hint: Use symmetry . )
Problem 5.R.102c
Function defined by an integral Let H (π) = β«βΛ£ β(4 β tΒ²) dt, for β 2 β€ π β€ 2.
(c) Evaluate H '(2) .
Problem 5.R.23d
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(d) β«ββ· Ζ(π) dπ
Problem 5.R.9a
Integration by Riemann sums Consider the integral β«ββ΄ (3πβ 2) dπ.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
Problem 5.R.55
Evaluating integrals Evaluate the following integrals.
β«βΒΉ π β’ 2Λ£Β²βΊΒΉ dπ
Problem 5.R.105c
Area functions and the Fundamental Theorem 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.
(c) Use the Fundamental Theorem to find an expression for F '(π) for 0 β€ π < 2.
Problem 5.R.95
Displacement from velocity A particle moves along a line with a velocity given by v(t) = 5 sin Οt, starting with an initial position s(0) = 0 . Find the displacement of the particle between t = 0 and t = 2 , which is given by s(t) = β«βΒ² v(t) dt . Find the distance traveled by the particle during this interval, which is β«βΒ² |v(t)| dt .
Problem 5.R.97
Find the average value of Ζ(π) = eΒ²Λ£ on [0, ln 2] .
Problem 5.R.41
Evaluating integrals Evaluate the following integrals.
β« (9πβΈβ7πβΆ) dπ
Problem 5.R.23b
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(b) β«ββ΄ Ζ(π) dπ
Problem 5.R.1a
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.
(a) A(π) = β«βΛ£ Ζ(t) dt and Ζ(t) = 2tβ3 , then A is a quadratic function.
Problem 5.R.45
Evaluating integrals Evaluate the following integrals.
β«Ο/β^Ο/Β³ (secΒ² t + cscΒ² t) dt
Problem 5.R.43
Evaluating integrals Evaluate the following integrals.
β«βΒΉ βπ (βπ + 1) dπ
Problem 5.R.46
Evaluating integrals Evaluate the following integrals.
β«Ο/ββ^Ο/βΉ (csc 3π cot 3π + sec 3π tan 3π) dπ
Problem 5.R.105d
Area functions and the Fundamental Theorem 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.
(d) Evaluate F ' (β1) and F ' (1). Interpret these values.
Problem 5.R.30
Evaluate the following derivatives.
d/dπ β«βα΅Λ£ cos tΒ² dt
![Graph of a function f(x) with a peak at (4,5) on the interval [2,6], showing axes labeled x and y.](https://static.studychannel-dev.pearsondev.tech/courses/calculus/thumbnails/644163eb-ecf8-4691-b0f1-f30360c250e2)
