Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Introduction to Definite Integrals

Calculus

8. Definite Integrals

Introduction to Definite Integrals

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3:03 minutes

Problem 5.R.1f

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(f) ∫ₐᵇ (2 ƒ(𝓍) ―3g (𝓍)) d𝓍 = 2 ∫ₐᵇ ƒ(𝓍) d𝓍 + 3 ∫₆ᵃ g(𝓍) d𝓍 .

Verified step by step guidance

Step 1: Begin by recalling the linearity property of definite integrals. This property states that for any continuous functions ƒ(𝓍) and g(𝓍), and constants c₁ and c₂, the integral of their linear combination can be expressed as: ∫ₐᵇ [c₁ƒ(𝓍) + c₂g(𝓍)] d𝓍 = c₁∫ₐᵇ ƒ(𝓍) d𝓍 + c₂∫ₐᵇ g(𝓍) d𝓍.

Step 2: Analyze the given statement: ∫ₐᵇ (2ƒ(𝓍) ― 3g(𝓍)) d𝓍 = 2∫ₐᵇ ƒ(𝓍) d𝓍 + 3∫₆ᵃ g(𝓍) d𝓍. Notice that the integral on the left-hand side involves a linear combination of ƒ(𝓍) and g(𝓍), which suggests the linearity property might apply.

Step 3: Compare the right-hand side of the equation. The term 2∫ₐᵇ ƒ(𝓍) d𝓍 is consistent with the linearity property. However, the term 3∫₆ᵃ g(𝓍) d𝓍 is problematic because the limits of integration are reversed (from 6 to a instead of a to b). Recall that reversing the limits of integration introduces a negative sign: ∫₆ᵃ g(𝓍) d𝓍 = -∫ₐ⁶ g(𝓍) d𝓍.

Step 4: Rewrite the right-hand side using the corrected limits of integration for g(𝓍). The equation should now read: ∫ₐᵇ (2ƒ(𝓍) ― 3g(𝓍)) d𝓍 = 2∫ₐᵇ ƒ(𝓍) d𝓍 - 3∫ₐ⁶ g(𝓍) d𝓍. This matches the linearity property of integrals.

Step 5: Conclude that the original statement is false because the limits of integration for g(𝓍) were incorrectly reversed in the given equation. The correct form of the equation should respect the linearity property and properly handle the limits of integration.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Definite Integrals

Definite integrals have specific properties that allow for the manipulation of integrals involving sums and constants. One key property is that the integral of a sum of functions can be expressed as the sum of their integrals. Additionally, constants can be factored out of the integral, which is crucial for simplifying expressions involving integrals.

Linearity of Integration

The linearity of integration states that for any two functions ƒ and g, and constants a and b, the integral of a linear combination of these functions can be expressed as the linear combination of their integrals. This means that ∫(aƒ(x) + b g(x)) dx = a∫ƒ(x) dx + b∫g(x) dx, which is essential for evaluating integrals involving multiple functions.

Continuity of Functions

Continuity of functions is a fundamental concept in calculus that ensures the behavior of functions is predictable over their domains. If ƒ and ƒ' are continuous, it implies that the functions do not have any breaks, jumps, or asymptotes, which is important for applying the Fundamental Theorem of Calculus and ensuring that the properties of integrals hold true.