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문제 5.R.96a
Velocity to displacement An object travels on the 𝓍-axis with a velocity given by v(t) = 2t + 5, for 0 ≤ t ≤ 4.
(a) How far does the object travel, for 0 ≤ t ≤ 4 ?
문제 5.R.1d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(d) If ƒ is continuous on [a,b] and ∫ₐᵇ |ƒ(𝓍)| d𝓍 = 0 , then ƒ(𝓍) = 0 on [a,b] .
문제 5.R.48
Evaluating integrals Evaluate the following integrals.
∫₁⁴ ((√v + v) / v ) dv
문제 5.R.18
Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.
―∫₄¹ 2ƒ(𝓍) d𝓍
문제 5.R.23a
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.
(a) ∫₀⁴ ƒ(𝓍) d𝓍
문제 5.R.23c
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.
(c) ∫₅⁷ ƒ(𝓍) d𝓍
문제 5.R.1g
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(g) ∫ ƒ' (g(𝓍))g' (𝓍) d(𝓍) = ƒ(g(𝓍)) + C .
문제 5.R.104
Change of variables Use the change of variables u³ = 𝓍² ― 1 to evaluate the integral ∫₁³ 𝓍∛(𝓍²―1) d𝓍 .
문제 5.R.102a
Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(a) Evaluate H (0) .
문제 5.R.25
Use geometry and properties of integrals to evaluate the following definite integrals.
∫₀⁴ √(8𝓍―𝓍²) d𝓍 . (Hint: Complete the square .)
문제 5.R.60
Evaluating integrals Evaluate the following integrals.
∫ sin 𝒵 sin (cos 𝒵) d𝒵
문제 5.R.81
Evaluating integrals Evaluate the following integrals.
∫√₂/₅^²/⁵ d𝓍/𝓍√(25𝓍² ―1)
문제 5.R.99b
(b) Find the average value of ƒ 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.
문제 5.R.9d
Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
문제 5.R.71
Evaluating integrals Evaluate the following integrals.
∫₋₅⁵ ω³ /√(ω⁵⁰ + ω²⁰ + 1) dω (Hint: Use symmetry . )
문제 5.R.102c
Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(c) Evaluate H '(2) .
문제 5.R.23d
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.
(d) ∫₀⁷ ƒ(𝓍) d𝓍
문제 5.R.9a
Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
문제 5.R.55
Evaluating integrals Evaluate the following integrals.
∫₀¹ 𝓍 • 2ˣ²⁺¹ d𝓍
문제 5.R.105c
Area functions and the Fundamental Theorem Consider the function
ƒ(t) = { t if ―2 ≤ t < 0
t²/2 if 0 ≤ t ≤ 2
and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.
(c) Use the Fundamental Theorem to find an expression for F '(𝓍) for 0 ≤ 𝓍 < 2.
문제 5.R.95
Displacement from velocity A particle moves along a line with a velocity given by v(t) = 5 sin πt, 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) = ∫₀² v(t) dt . Find the distance traveled by the particle during this interval, which is ∫₀² |v(t)| dt .
문제 5.R.97
Find the average value of ƒ(𝓍) = e²ˣ on [0, ln 2] .
문제 5.R.41
Evaluating integrals Evaluate the following integrals.
∫ (9𝓍⁸―7𝓍⁶) d𝓍
문제 5.R.23b
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.
(b) ∫₆⁴ ƒ(𝓍) d𝓍
문제 5.R.1a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(a) A(𝓍) = ∫ₐˣ ƒ(t) dt and ƒ(t) = 2t―3 , then A is a quadratic function.
문제 5.R.45
Evaluating integrals Evaluate the following integrals.
∫π/₆^π/³ (sec² t + csc² t) dt
문제 5.R.43
Evaluating integrals Evaluate the following integrals.
∫₀¹ √𝓍 (√𝓍 + 1) d𝓍
문제 5.R.46
Evaluating integrals Evaluate the following integrals.
∫π/₁₂^π/⁹ (csc 3𝓍 cot 3𝓍 + sec 3𝓍 tan 3𝓍) d𝓍
문제 5.R.105d
Area functions and the Fundamental Theorem Consider the function
ƒ(t) = { t if ―2 ≤ t < 0
t²/2 if 0 ≤ t ≤ 2
and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.
(d) Evaluate F ' (―1) and F ' (1). Interpret these values.
문제 5.R.30
Evaluate the following derivatives.
d/d𝓍 ∫₃ᵉˣ cos t² 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.pearsonprd.tech/courses/calculus/thumbnails/644163eb-ecf8-4691-b0f1-f30360c250e2)
