8. Definite Integrals

Evaluate the integrals in Exercises 47–68.

∫₁² 4 dv

v²

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.

a. ∫²₋₂ ƒ(x) dx

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

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.

e. ∫⁵₋₂ ( ƒ(x) + g(x) ) dx

5

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

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

Evaluate the integrals in Exercises 23–32.

∫₀^π √(1 - cos(2x)) dx

Evaluate the integrals in Exercises 23–32.

∫₀^π √(1 - cos²(θ)) dθ

Evaluate the integrals in Exercises 53–58.

∫ from 0 to π/2 of sin(x) cos(x) dx

Evaluate the integrals in Exercises 53–58.

∫ from -π/2 to π/2 of cos(x) cos(7x) dx

Evaluate the limits in Exercise 7 and 8.

lim (x → ∞) ∫₋ˣ^ˣ sin t dt

Evaluate the integrals in Exercises 97–110.

99. ∫₀³ (√2 + 1)x^(√2) dx

Evaluate the integrals in Exercises 41–60.

59. ∫(from -ln2 to 0)cosh²(x/2) dx

Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year’s rainfall will exceed 17 in.?

Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.

a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?