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Problem 5.PE.39
Evaluate the integrals in Exercises 37–46.
∫(2θ + 1 + 2 cos (2θ + 1))dθ
Problem 5.PE.82
Find dy/dx if y = ∫(From cos x to 0) 1/(1 - t²) dt.
Explain the main steps in your calculation.
Problem 5.PE.9d
If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.
d. ∫⁵₋₂ (-πg(x)) dx
Problem 5.PE.21
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y² = 4x, y = 4x - 2
Problem 5.PE.68
Evaluate the integrals in Exercises 47–68.
∫₀^π/4 sec²x / (1 + 7 tan x)²/³ dx
Problem 5.PE.10d
If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.
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d. ∫₀² √2ƒ(x) dx
Problem 5.PE.27
Find the area of the “triangular” region bounded on the left by x + y = 2, on the right by y = x², and above by y = 2.
Problem 5.PE.56
Evaluate the integrals in Exercises 47–68.
∫₀¹/² x³ (1 + 9x⁴)⁻³/² dx
Problem 5.PE.63
Evaluate the integrals in Exercises 47–68.
∫⁰-π/3 sec x tan x dx
Problem 5.PE.7
Definite Integrals
In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.
n
lim ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0]
∥P∥→0 k = 1
Problem 5.PE.3a
10 10
Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of
k = 1 k = 1
10
a. Σ aₖ/4
k = 1
Problem 5.PE.62
Evaluate the integrals in Exercises 47–68.
∫₀ ^π tan² (θ/3) dθ
Problem 5.PE.49
Evaluate the integrals in Exercises 47–68.
∫₁² 4 dv
v²
Problem 5.PE.47
Evaluating Definite Integrals
Evaluate the integrals in Exercises 47–68.
∫₋₁¹ (3x² - 4x + 7)dx
Problem 5.PE.11
Area
In Exercises 11–14, find the total area of the region between the graph of ƒ and the x-axis.
ƒ(x) = x² - 4x + 3, 0 ≤ x ≤ 3
Problem 5.PE.70a
Find the average value of
__
a. y = √3x over [0, 3]
Problem 5.PE.75
Differentiating Integrals
In Exercises 75–78, find dy/dx.
________
y = ∫₂ˣ √ 2 + cos³t dt
Problem 5.PE.3c
10 10
Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of
k = 1 k = 1
10
c. Σ (aₖ + bₖ - 1)
k = 1
Problem 5.PE.59
Evaluate the integrals in Exercises 47–68.
∫₀^π/3 sec² θ dθ
Problem 5.PE.52
Evaluate the integrals in Exercises 47–68.
_
∫₁⁴ (1 + √u)¹/² du
√u
Problem 5.PE.17
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
√x + √y = 1, x = 0, y = 0
Problem 5.PE.4a
20 20
Suppose that Σ aₖ = 0 and Σ bₖ = 7. Find the value of
k = 1 k = 1
20
a. Σ 3aₖ
k = 1
Problem 5.PE.5
Definite Integrals
In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.
n
lim ∑ (2cₖ - 1)⁻¹/² ∆xₖ, where P is a partition of [1, 5]
∥P∥→0 k = 1
Problem 5.PE.31
Find the total area of the region enclosed by the curve x = y²/³ and the lines x = y and y = -1.
Problem 5.PE.10b
If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.
b. ∫₁² g(x) dx
Problem 5.PE.77
In Exercises 75–78, find dy/dx.
y = ∫(from x to 1) (6/(3 + t^4))dt
Problem 5.PE.25
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = 2 sin x, y = sin 2x, 0 ≤ x ≤ π
Problem 5.PE.23
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = sin x, y = x, 0 ≤ x ≤ π/4
Problem 5.PE.13
In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.
ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8
Problem 5.PE.15
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = x, y = 1/x², x = 2
