Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍...

Question

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

∫₁³ ƒ(𝓍)/g(𝓍) d𝓍

∫₁³ ƒ(𝓍)/g(𝓍) d𝓍

Accepted Answer

Welcome back, everyone. Given the integral from 0 to 3 of PFXDX equals 8, the integral from 0 to 1 of QXDX equals 5, and the integral from 1 to 3 of PF XDX equals 6 determine the value of the integral from 0 to 1 of PX X divided by QXDX or state that there is not enough information. AA 13, B8. C 3 and there is not enough information. So, for this problem, we want to evaluate the integral from 0 to 1 of P X divided by Q XD X. We are given the integrals of PF X and Q of X with different limits of integration, right? First of all, let's notice that we have the integral of Q of XDX with the same limits of integration from 0 to 1, and this integral is equal to 5. Now what we can do is simply solve for the integral from 0 to 1 of PXDX. How do we do that? Well, we are going to use two integrals. We are going to take the integral from 0 to 3. Using the properties of integrals for the whole interval from 0 to 3 of PFXDX. And we are going to subtract the integral from 1 to 3 of PFXDX. This essentially allows us to identify the integral of PF XDX using the properties of definite integrals, right? This is going to be equal to 8 because the value of the first integral is 8. -6. Which is equal to 2. Now let's notice that now we have the integrals from 0 to 1 of both BFX and Q of X, right? So essentially, we want to identify the integral of the ratio of the two functions. Let's remember that there is no property which allows us to integrate the ratio. Of PFX divided by Q of X, as the ratio of two separate integrals, right? So, this integral is not equal to the integral from 0 to 1 of PFXDX. Divided by the integral from 0 to 1 of Q of XDX. So there is no property that allows us to split the integral of a ratio into a ratio of two separate integrals, right? Because there is no such property, we can conclude that there is not enough information. We must know the exact function PFX and the exact function. Q of X, right? And then we must integrate that ratio under one integral, meaning the correct answer to this problem is the, there is not enough information. Thank you for watching.

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.∫₁³ ƒ(𝓍)/g(𝓍) d𝓍

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Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

∫₁³ ƒ(𝓍)/g(𝓍) d𝓍