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

Previous problemNext problem

1:52 minutes

Problem 5.2.69d

Textbook Question

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.

Verified step by step guidance

Step 1: Begin by understanding the integral notation. The expression ∫ₐᵇ ƒ(𝓍) d𝓍 represents the definite integral of the function ƒ(𝓍) over the interval [a, b]. This computes the net area under the curve of ƒ(𝓍) between x = a and x = b.

Step 2: Analyze the given statement. The statement claims that if ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ must be a constant function. This implies that the equality of the integrals is being used to infer something about the nature of the function ƒ.

Step 3: Consider whether the equality of the integrals necessarily implies that ƒ is constant. Recall that the definite integral depends on the values of ƒ(𝓍) over the interval [a, b], but it does not directly indicate whether ƒ is constant. For example, two different functions can have the same integral value over the same interval.

Step 4: Provide a counterexample to disprove the statement if necessary. For instance, consider two functions ƒ₁(𝓍) = 𝓍 and ƒ₂(𝓍) = 𝓍² - 𝓍 over a specific interval [a, b]. Compute their integrals and observe that they may yield the same result, even though neither function is constant.

Step 5: Conclude that the statement is false. The equality of the integrals does not necessarily imply that ƒ is a constant function. The integral only provides information about the net area under the curve, not the specific behavior of the function across the interval.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The value of a definite integral can provide insights into the behavior of the function, such as total accumulation or net change over the interval.

Constant Function

A constant function is a function that always returns the same value regardless of the input variable. Mathematically, it can be expressed as f(x) = c, where c is a constant. In the context of integrals, if the integral of a function over an interval is equal to itself, it does not necessarily imply that the function is constant; rather, it indicates that the area under the curve remains unchanged.

Counterexample

A counterexample is a specific case that disproves a general statement or proposition. In calculus, providing a counterexample can effectively demonstrate that a certain condition does not hold true for all functions. For instance, if two integrals are equal, one can find a non-constant function that satisfies this equality, thus serving as a counterexample to the claim that the function must be constant.

Previous problemNext problem

5:43m