Back
Problem 5.4.23
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫²₋₂ [(x³ ― 4x) / (x² + 1)] dx
Problem 5.3.84
Derivatives of integrals Simplify the following expressions.
d/dt ∫₀ᵗ d𝓍/(1 + 𝓍²) + ∫₁¹/ᵗ dx/(1 + 𝓍²)
Problem 5.1.61
{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.
The right Riemann sum for ƒ(𝓍)) = x + 1 on [0, 4] with n = 50.
Problem 5.2.35
Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function ƒ on [a,b]. Identify ƒ and express the limit as a definite integral.
n
lim ∑ (𝓍ₖ*² + 1) ∆𝓍ₖ on [0,2]
∆ → 0 k=1
Problem 5.2.39
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.
∫₀⁴ (8―2𝓍) d𝓍
Problem 5.3.5
The linear function ƒ(𝓍) = 3 ― 𝓍 is decreasing on the interval [0, 3]. Is its area function for ƒ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.
Problem 5.5.74
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₀^π/⁴ eˢᶦⁿ² ˣ sin 2𝓍 d𝓍
Problem 5.5.40
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ (sin⁵ 𝓍 + 3 sin³ 𝓍― sin 𝓍) cos 𝓍 d𝓍
Problem 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
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 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = π
Problem 5.5.80
Variations on the substitution method Evaluate the following integrals.
∫ y²/(y + 1)⁴ dy
Problem 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
Problem 5.4.33
Average distance on a parabola What is the average distance between the parabola y = 30𝓍 (20 ― 𝓍 ) and the 𝓍-axis on the interval [0, 20] ?
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 ≤ t ≤ 8 ; n = 4
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.
∫₀¹ (𝓍² ― 2𝓍 + 3) d𝓍
Problem 5.5.36
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ sec 4w tan 4w dw
Problem 5.4.56
Average value of the derivative Suppose ƒ ' is a continuous function for all real numbers. Show that the average value of the derivative on an interval [a, b] is ƒ⁻' = (ƒ(b) ―ƒ(a))/ (b―a) . Interpret this result in terms of secant lines.
Problem 5.5.12
Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals
∫(e³ˣ ⁺¹ d𝓍
Problem 5.2.81
Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.
∫₃⁷ (4𝓍 + 6) d𝓍
Problem 5.3.31
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
∫₁⁸ 8𝓍¹/³ d𝓍
Problem 5.4.49
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
∫ᵃ₋ₐ ƒ(g(𝓍)) d𝓍
Problem 5.5.46
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₀¹ 2e²ˣ d𝓍
Problem 5.5.92
Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.
∫ 𝓍 cos²𝓍² d𝓍
Problem 5.3.114
Max/min of area functions Suppose ƒ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of ƒ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of ƒ. Verify this fact with the function ƒ(𝓍) = 𝓍² - 10𝓍.
Problem 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]
Problem 5.5.10
Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.
∫ (6𝓍 + 1) √(3𝓍² + 𝓍) d𝓍 , u = 3𝓍² + 𝓍
Problem 5.1.67
Identifying Riemann sums Fill in the blanks with an interval and a value of n.
4
∑ ƒ (1.5 + k) • 1 is a midpoint Riemann sum for f on the interval [ ___ , ___ ]
k = 1
with n = ________ .
Problem 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
Problem 5.3.8
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Problem 5.2.59
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.
∫₀ᵃ ƒ(𝓍) d𝓍
