Indietro
Problema 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 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = π
Problema 5.5.20
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍
Problema 5.4.18
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫₋π/₂^π/² 5 sin θ dθ
Problema 5.3.2
Suppose F is an antiderivative of ƒ and A is an area function of ƒ. What is the relationship between F and A?
Problema 5.5.43
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ sin 𝓍 sec⁸ 𝓍 d𝓍
Problema 5.5.32
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ 2 / (𝓍√4𝓍² ―1) d𝓍 , 𝓍 > ½
Problema 5.5.116
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
∫ 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)
Problema 5.5.70
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₋₁¹ (𝓍―1) (𝓍²―2𝓍)⁷ d𝓍
Problema 5.5.1
On which derivative rule is the Substitution Rule based?
Problema 5.5.106
General results Evaluate the following integrals in which the function ƒ is unspecified. Note that ƒ⁽ᵖ⁾ is the pth derivative of ƒ and ƒᵖ is the pth power of ƒ. Assume ƒ and its derivatives are continuous for all real numbers.
∫ (5 ƒ³ (𝓍) + 7ƒ² (𝓍) + ƒ (𝓍 )) ƒ'(𝓍) d𝓍
Problema 5.3.106
{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of ƒ and the 𝓍-axis on the given interval. Graph ƒ and show the region 𝑅.
ƒ(𝓍) = 𝓍² (𝓍 ― 2) on [ ―1 , 3]
Problema 5.3.103
{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of ƒ and the 𝓍-axis on the given interval. Graph ƒ and show the region 𝑅.
ƒ(𝓍) = 2 ― |𝓍| on [ ― 2 , 4]
Problema 5.2.61
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.
∫ₐᶜ ƒ(𝓍) d𝓍
Problema 5.5.84
Variations on the substitution method Evaluate the following integrals.
∫ (𝒵 + 1) √(3𝒵 + 2) d𝒵
Problema 5.5.117
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
∫ d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)
Problema 5.5.59
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₂/₍₅√₃₎^²/⁵ d𝓍/ x√(25𝓍²― 1)
Problema 5.3.8
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Problema 5.3.93
Area functions from graphs The graph of ƒ is given in the figure. A(𝓍) = ∫₀ˣ ƒ(t) dt and evaluate A(2), A(5), A(8), and A(12).
Problema 5.5.35
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ 𝓍 csc 𝓍² cot 𝓍² d𝓍
Problema 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.
∫₀² (2𝓍 + 1) d𝓍
Problema 5.3.10
Explain why ∫ₐᵇ ƒ ′(𝓍) d𝓍 = ƒ(b) ― ƒ(a)
Problema 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 ≤ t ≤ 3.
v(t) = { 30 if 0 ≤ t ≤ 2
50 if 2 < t < 2.5
44 if 2.5 < t ≤ 3
Problema 5.2.87
Area by geometry Use geometry to evaluate the following integrals.
∫⁴₋₆ √(24 ― 2𝓍 ― 𝓍²) d𝓍
Problema 5.1.23
Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n.
ƒ(𝓍) = x + 1 on [1,6] ; n = 5
Problema 5.4.15
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫²₋₂ (x² + x³) dx
Problema 5.3.100
Determine the intervals on which the function g(𝓍) = ∫ₓ⁰ t / (t² + 1) dt is concave up or concave down.
Problema 5.5.87
Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.
∫₋π^π cos² 𝓍 d𝓍
Problema 5.5.37
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ sec² (10𝓍 + 7) d𝓍
Problema 5.5.22
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ [ 1/(10𝓍―3) d𝓍
Problema 5.2.67
Use geometry and properties of integrals to evaluate
∫₀¹ (2𝓍 + √(1―𝓍²) + 1) d𝓍
