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Problem 5.3.51
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«ββ΄ (π β 2)/βπ 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 5.5.88
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β« sinΒ² π dπ
Problem 5.5.80
Variations on the substitution method Evaluate the following integrals.
β« yΒ²/(y + 1)β΄ dy
Problem 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.
β« π csc πΒ² cot πΒ² dπ
Problem 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.
β« [(βπ + 1)β΄ / 2βπ dπ
Problem 5.4.31
Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
Ζ(π) = πβΏ on [0,1] , for any positive integer n
Problem 5.5.59
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«β/ββ βββ^Β²/β΅ dπ/ xβ(25πΒ²β 1)
Problem 5.5.63
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«β/β^ΒΉ/βΒ³ 4/(9πΒ² + 1) dπ
Problem 5.3.11
Evaluate β«ββΈ Ζ β²(t) dt , where Ζ β² is continuous on [3, 8], Ζ(3) = 4, and Ζ(8) = 20 .
Problem 5.3.35
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«ββΉ 2/(βπ) dπ
Problem 5.4.25
Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
Ζ(π) = πΒ³ on [β1, 1]
Problem 5.2.87
Area by geometry Use geometry to evaluate the following integrals.
β«β΄ββ β(24 β 2π β πΒ²) dπ
Problem 5.5.11
Use a substitution of the form u = aπ + b to evaluate the following indefinite integrals.
β«(π + 1)ΒΉΒ² dπ
Problem 5.1.21
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.
{Use of Tech} v = 4 β(t +1) (mi/hr) . for 0 β€ t β€ 15 ; n = 5
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.
β«ββ· (4π + 6) dπ
Problem 5.3.81
Derivatives of integrals Simplify the following expressions.
d/dz β«ΒΉβ°βα΅’β β dt /(tβ΄ + 1)
Problem 5.5.66
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«βα΅Β² (ln p)/p dp
Problem 5.4.38
Average height of a wave The surface of a water wave is described by y = 5 (1 + cos π) , for β Ο β€ π β€ Ο, where y = 0 corresponds to a trough of the wave (see figure). Find the average height of the wave above the trough on [ βΟ , Ο] .
Problem 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.
β«βΒ² (2π + 1) dπ
Problem 5.5.3
The composite function Ζ(g(π)) consists of an inner function g and an outer function Ζ. If an integrand includes Ζ(g(π)), which function is often a likely choice for a new variable u?
Problem 5.2.37
Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function Ζ on [a,b]. Identify Ζ and express the limit as a definite integral.
n
lim β π*β (ln π*β) βπβ on [1,2]
β β 0 k=1
Problem 5.1.63
{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.β
The midpoint Riemann sum for f(x) = xΒ³ on [3,11] with n = 32.
Problem 5.4.21
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«βΟ/β^Ο/β΄ secΒ² x dx
Problem 5.5.1
On which derivative rule is the Substitution Rule based?
Problem 5.4.35
Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = tΒ² + 3t. Find the average velocity of the object over this time interval.
Problem 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.
β« [ 1/(10πβ3) dπ
Problem 5.2.83
Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.
β«ββ΄ (πΒ²β1) dπ
Problem 5.4.5
Use symmetry to explain why.
β«β΄ββ (5πβ΄ + 3πΒ³ + 2πΒ² + π + 1) dπ = 2 β«ββ΄ (5πβ΄ + 2πΒ² + π + 1) dπ .
Problem 5.5.74
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«β^Ο/β΄ eΛ’αΆ¦βΏΒ² Λ£ sin 2π dπ
