8. Definite Integrals

Evaluate the integrals in Exercises 23–32.

∫₀^π √(1 - cos(2x)) dx

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

52. ∫ from 0 to π/2 of cos⁶x dx

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n. 

Evaluate the integrals in Exercises 39–56.

39. ∫(from -3 to -2)dx/x

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(a) ∫₀³ 5ƒ(𝓍) d𝓍

Find ${\int }_0^{6}f\left(x\right)dx$ given the graph of $y=f\left(x\right)$.

114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).

c. As a increases, L(a) increases as what power of a?

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

58. ∫₀^{2π} dt / (4 + 2 sin t)²

Area by geometry Use geometry to evaluate the following integrals.

∫⁴₋₆ √(24 ― 2𝓍 ― 𝓍²) d𝓍

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₄^π/⁴ sec² x dx

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(e) Find the value of s such that H (𝓍) = sH(―𝓍)

Evaluate the integrals in Exercises 53–58.

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

Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.

(f) ∫₄¹ 2f(𝓍) d𝓍

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₂^π/² 5 sin θ dθ

Evaluating integrals Evaluate the following integrals.

∫₀⁵ |2𝓍―8|d𝓍