Find the area of the following regions, expressing your result...

Question

Find the area of the following regions, expressing your results in terms of the positive integer n≥2.

Let Aₙ be the area of the region bounded by f(x)=x^1/n and g(x)=x^n on the interval [0,1], where n is a positive integer. Evaluate lim n→∞ Aₙ and interpret the result. br

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says let BM be the area of the region bounded by F of X, which is equal to X raised to the quantity of 2 divided by M in quantity and G of X be equal to X raised to the power M on the closed interval from 0 to 1, where M is a positive. Of integer with M being greater than or equal to 2. Compute the limit as M approaches infinity of BM, and we're given 4 possible choices as our answers. For choice A, we have 1 divided by 2. For choice B, we have 1, for choice C, we have 3 divided by 2, and for choice D, we have 4. So we're asked to compute the limit as M approaches infinity of BM, and we're told that BM is the area of the region bounded by F of X and G of X on the closed interval from 0 to 1. So that means that we can can compute this area through integration. So, BM is going to be equal to the integral from 0 to 1, since that is the info that we're looking at for this problem. Of the quantity, and here we have to determine which of our two functions forms the upper bound of the region, and which one forms the lower bound of our region. And we're going to be interested in the case when M is very large, since we ultimately want to take the limit as M approaches infinity. So if we look at F of X, which is equal to X, agrees to the quantity of 2 divided by M in quantity, we see that its exponent is going to be small, since. 2 divided by M is going to approach 0 as M approaches infinity. And if we look at G of X, which is equal to X-rays to the power M, its exponent is going to be very large as M approaches infinity. Now, since X has to be contained in the interval from 0 to 1, that means that raising X to a large power actually makes it smaller. So that means that G of X is actually going to be smaller than F of X. So here for integrand, we want to take the curve that is bounded above, minus the curve that bounds our region from below. And so, the curve that bounds our region from above is going to be F of X, which is gonna be X raised to the quantity of 2 divided by M in quantity, then we'll have minus. G of X, which is going to be X raised to the power M in quantity DX. So now that we have the integral set up, we can just calculate our integral by integrating term by term. In doing so, this is going to be equal to. The quantity and here when we integrate x raised to the quantity of 2 divided by m in quantity, we get X raised to the quantity of 2 divided by m plus 1 in quantity, and that is divided by the quantity of 2 divided by m plus 1 in quantity. Then we'll have minus and here we integrate. X raises to the power M. With respect to X, we get X raised to the quantity of m + 1 in quantity divided by the quantity of m + 1 in quantity, and this needs to be evaluated from 0 to 1. So here we just use our power rule to do the integration. So now we need to substitute in our limits of integration, so this is going to be equal to. 1 raised to the quantity of 2 divided by M plus 1 in quantity, divided by the quantity of 2 divided by M plus 1 in quantity, then we'll have minus 1 raised to the quantity of m plus 1 in quantity divided by the quantity of M + 1. Then we'll have minus. 0 raised to the quantity of 2 divided by m plus 1 in quantity, divided by the quantity of 2 divided by m plus 1 in quantity. Then we'll have plus 0 raised to the quantity of m plus 1 in quantity divided by the quantity of M + 1 in quantity. So now we need to simplify this expression. So this is going to be equal to. When we have 1 raise to any power, we're going to get 1, and so we'll have that divided by the quantity of 2 divided by M plus 1 in quantity, and then we'll have minus 1 divided by the quantity of M plus 1. In quanti For the next two terms, we do have to be careful since 0 being raised to the power 0 is an indeterminate form. But if we look at our exponents here, we have the quantity of 2 divided by M plus 1 in quantity for the first term exponents, and for the second term exponent, we have the quantity of M + 1. Now, since M has to be greater than or equal to 2, both of these exponents are positive integers that are greater than. 0. So that means that the last two terms are going to be 0. So we're gonna have minus 0 and then plus 0. And so, our final answer for BM is going to be equal to 1 divided by the quantity of 2 divided by M plus 1 in quantity minus 1 divided by the quantity of M plus 1 in quantity. So now we want to take the limit as M approaches infinity of this expression. So we'll have the limit. As M approaches infinity of BM, and this is going to be equal to the limit, as M approaches infinity. Of the quantity of 1 divided by the quantity of 2 divided by M. Plus 1 in quantity minus 1 divided by the quantity of m plus 1 in quantity. So, as M approaches infinity, the quantity of 2 divided by M is going to approach 0. And the quantity of 1, divided by the quantity of M + 1 and quantity is also going to approach 0. So evaluating this limit, we see that This is going to be equal to 1, divided by the quantity of 0 plus 1 in quantity minus 0. And simplifying this expression, we see that this limit is going to be equal to 1, and that is the answer that we're looking for for this problem. And if we compare our answer to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Find the area of the following regions, expressing your results in terms of the positive integer n≥2.Let Aₙ be the area of the region bounded by f(x)=x^1/n and g(x)=x^n on the interval [0,1], where n is a positive integer. Evaluate lim n→∞ Aₙ and interpret the result. br

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Find the area of the following regions, expressing your results in terms of the positive integer n≥2.

Let Aₙ be the area of the region bounded by f(x)=x^1/n and g(x)=x^n on the interval [0,1], where n is a positive integer. Evaluate lim n→∞ Aₙ and interpret the result. br