Textbook Question
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
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8. Definite Integrals
Introduction to Definite Integrals
2:54 minutesProblem 5.2.55f
Textbook Question
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
Verified step by step guidance1
Step 1: Recognize that the integral ∫₄¹ 2f(𝓍) d𝓍 involves a constant multiplier. Use the property of integrals that states ∫ₐᵇ c·ƒ(𝓍) d𝓍 = c·∫ₐᵇ ƒ(𝓍) d𝓍, where c is a constant.
Step 2: Rewrite the integral as 2·∫₄¹ ƒ(𝓍) d𝓍 using the property mentioned in Step 1.
Step 3: Notice that the limits of integration are reversed (from 4 to 1 instead of 1 to 4). Use the property of integrals that states ∫ₐᵇ ƒ(𝓍) d𝓍 = -∫ₐᵇ ƒ(𝓍) d𝓍 when the limits are swapped.
Step 4: Apply the property from Step 3 to rewrite the integral as -2·∫₁⁴ ƒ(𝓍) d𝓍.
Step 5: Use the given information that ∫₁⁶ ƒ(𝓍) d𝓍 = 10 and ∫₄⁶ ƒ(𝓍) d𝓍 = 5 to deduce that ∫₁⁴ ƒ(𝓍) d𝓍 = ∫₁⁶ ƒ(𝓍) d𝓍 - ∫₄⁶ ƒ(𝓍) d𝓍. Substitute this value into the expression from Step 4 to complete the setup for evaluation.
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Key Concepts
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