Textbook Question
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
9
views
8. Definite Integrals
Introduction to Definite Integrals
1:43 minutesProblem 5.2.55b
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.
(b) ∫₁⁶ (f(𝓍) ― g(𝓍)) d𝓍
Verified step by step guidance1
Step 1: Recall the property of integrals that states ∫ₐᵇ (ƒ(𝓍) ± g(𝓍)) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍 ± ∫ₐᵇ g(𝓍) d𝓍. This allows us to split the integral into two separate integrals.
Step 2: Apply the property to the given integral ∫₁⁶ (ƒ(𝓍) ― g(𝓍)) d𝓍. This becomes ∫₁⁶ ƒ(𝓍) d𝓍 ― ∫₁⁶ g(𝓍) d𝓍.
Step 3: Substitute the given values for the integrals: ∫₁⁶ ƒ(𝓍) d𝓍 = 10 and ∫₁⁶ g(𝓍) d𝓍 = 5.
Step 4: Perform the subtraction operation symbolically: 10 ― 5.
Step 5: The result of the subtraction gives the value of the integral ∫₁⁶ (ƒ(𝓍) ― g(𝓍)) d𝓍. This completes the evaluation process.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Definite Integrals
Definite integrals have several key properties, including linearity, which states that the integral of a sum of functions is the sum of their integrals. This means that for any two functions f and g, ∫(f + g) dx = ∫f dx + ∫g dx. Additionally, the integral of a constant multiplied by a function can be factored out: ∫k * f dx = k * ∫f dx. Understanding these properties is essential for evaluating integrals involving combinations of functions.
Recommended video:
05:43
Definition of the Definite Integral
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫ₐᵇ f(x) dx = F(b) - F(a). This theorem allows us to evaluate definite integrals by finding the antiderivative of the integrand. It is crucial for solving problems involving definite integrals, as it provides a method to compute the area under a curve.
Recommended video:
06:11Fundamental Theorem of Calculus Part 1
Substitution in Integrals
Substitution is a technique used in integration to simplify the process of finding integrals. It involves changing the variable of integration to make the integral easier to evaluate. For example, if we let u = g(x), then the integral ∫f(g(x))g'(x)dx can be transformed into ∫f(u)du. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in a simpler form.
Recommended video:
04:27Substitution With an Extra Variable
5:43mWatch next
Master Definition of the Definite Integral with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice