Properties of integrals Consider two function...

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.

(d) ∫₄⁶ (g(𝓍) ― f(𝓍) d𝓍

(d) ∫₄⁶ (g(𝓍) ― f(𝓍) d𝓍

Accepted Answer

Welcome back, everyone. Suppose P and Q are functions on the interval from 0 to 4 inclusive, such that the integral from 0 to 4 of QXDX is equal to 9, the integral from 2 to 4 of PF XDX is equal to 3, and the integral from 0 to 2 of QXDX is equal to 4. Find the integral from 2 to 4 of Q X minus PXDX. A -2, B -8, C 10, and D2. So let's begin by considering the integral that we want to compute, that would be the integral from 2 to 4. Of Q X minus p of XD X. And we can apply the properties of integrals to rewrite it as the difference of two integrals, right? So, the integral from 2 to 4 of our first function Q of XD X. Minus the integral from 2 to 4 of our second function p of X. D X Now what we have to consider is the initial data given to us in the problem. Let's consider the first integral from 2 to 4 of Q of XDX. We are not provided with the value of this integral. However, we have the integral from 2 to 4 of PXDX. It is equal to 3, right? So we can label our second integral as 3, right? So, we're subtracting 3. And now our goal is to consider the integral from 2 to 4 of Q of XDX. We can notice that in the problem we're given the integral of QXDX from 0 to 4 and from 0 to 2, so we can essentially apply one of the properties of integrals and we can say that the integral from 0 to 4 of Q of XDX. Can be written as a sum of two integrals of Q of XD X. And we're going to consider our limits of integration, so that would be from 0 to 2. And then from 2 to 4. Right? Now, we know that the integral from 0 to 4 of Q of XDX is 9, so we're going to label the left hand side. S 9. On the right hand side, we have the integral from 0 to 2, which is 4. And then the integral from 2 to 4 is unknown. So, in order to identify its value, we have to subtract 9 minus 4, which leaves us with 5. Meaning the integral from 2 to 4 of QXDX is 5, and we can now compute the value of the integral from 2 to 4 of Q X minus PXXDX as 5 minus 3. Which is equal to 2. Therefore, the correct answer to this problem corresponds to the answer choice d. Thank you for watching.

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

Key Concepts

Properties of Definite Integrals

Fundamental Theorem of Calculus

Integration of Functions over Intervals

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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

Key Concepts

Properties of Definite Integrals

Fundamental Theorem of Calculus

Integration of Functions over Intervals

Watch next

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