8. Definite Integrals

Evaluate ∫₀² 3𝓍² d𝓍 and ∫₋₂² 3𝓍² d𝓍. 

Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?

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

(c) ∫₃⁶ (3ƒ(𝓍) ― g(𝓍)) d𝓍

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

(d) ∫₆³ (ƒ(𝓍) + 2g(𝓍)) 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.

(a) ∫₁⁴ 3f(𝓍) 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.

(b) ∫₁⁶ (f(𝓍) ― g(𝓍)) 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.

(d) ∫₄⁶ (g(𝓍) ― f(𝓍) 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

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀¹ (𝓍³ ― 2𝓍) d𝓍 = ―3/4

(a) ∫₀¹ (4𝓍―2𝓍³) d𝓍

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀¹ (𝓍³ ― 2𝓍) d𝓍 = ―3/4

(b) ∫₁⁰ (2𝓍―𝓍³) d𝓍

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1

(a) ∫₀^π/2 (2 sin θ ― cos θ) dθ

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1

(b) ∫₀^π/2 (4 cos θ ― 8 sin θ) dθ

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

(d) If ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ is a constant function. 

75. Exploring powers of sine and cosine

c. Prove that ∫₀ᵖⁱ sin²(nx) dx has the same value for all positive integers n.

75. Exploring powers of sine and cosine

e. Repeat parts (a), (b), and (c) with sin²x replaced by sin⁴x. Comment on your observations.