The following functions are positive and negative on the given...

Question

The following functions are positive and negative on the given interval.

ƒ(𝓍) = xe⁻ˣ on [-1,1]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

Accepted Answer

Hello. In this video, we are given a function F of X, which is equal to 1/2 X, multiplied by E, raised to the power of -2 X. And we are told that we are working with this function on the boundary from -1 to 1. By using N is equal to 4, what is the right to rhyme in some approximation for the net area bounded by the graph of F on and the X-axis on the interval? So, in order to approach this problem, we are going to be using the definition of the definite integral. We are the definite integral is defined as the integral from A to B of a function of X. And that is going to be equal to an infinite summation from 1 ton. Of F of X sub I multiplied by the changing X delta X. Here, delta X is defined as our boundaries B minus A divided by N, and Xi is defined as A plus I, which is the index multiplied by delta X. So, in this case here, what we need to do is we need to go ahead and first define our change in X. Delta X is going to be the difference of the boundaries. Now, in this case here, we are told that we are working on the boundary from -1 to 1. So B is going to equal to 1 and A is going to equal to -1. And this is going to be defined by our given value of N, which in this case is going to be 4. This will simplify to 1 + 1 divided by 4, which is going to give us 2/4, which is equivalent to 1/2. So, this is going to be the delta X that we are working with in the summation. Now what we want to do is we want to go ahead and express our function in a summation. So we're going to have the summation from Is equal to 1 ton. Of F of Xi. Delta X. Now, in this case, we are going to work with N is equal to 4. So we're going to replace N in the summation with 4. So this is telling us that we are taking the sum of the 1st 4 elements of F of X sub I. Now, this is going to simplify to be Delta X multiplied by each endpoint plugged into our function. So this is going to be FX1. Plus F of X 2. Plus F of X sub 3. Plus F of X 4. And now what we need to do is we need to define the end points that we are going to plug into the given function. Now, in this problem, we are told that we are working with a right to rhyme in sums, which means that we are going to need to define our right endpoints. So let's go ahead and create a number line to define these endpoints. Now this number line is going to be from -1 to 1. Since we are working with N equal to 4, we want to divide this area into 4 equal parts. And that means that the length of each part on the number line is going to have a total length of 14th. So, the next number to the right of -1 is going to be -5, 0, and. And what we want to do is we want to pick the right endpoints of this number line, which is going to be -5, 0, 1/2, and 1. So that is going to change. Our summation to be delta X, which is 1/2 multiplied by F-12 plus F of 0, plus F of 1/2 plus F of 1. And what this means is that we need to take each endpoint and plug them into the given function. Now, again, the function given to us is F of X is equal to 1/2 X multiplied by E rates to the negative to X. If we were to plug in. The first endpoint, which is -5, we are going to get the value of -14 E. Plugging in 0 into the function will give us 0, plugging in positive 1/2 will give us 1 divided by 4E, and plugging in the last N.1 is going to give us 1 divided by 2 E squared. And with the help of a calculator, this is going to approximate to be negative 0.260, which is going to be the approximation of the net area bounded between the function and the X axis. And with that being said, the solution to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

The following functions are positive and negative on the given interval.
ƒ(𝓍) = xe⁻ˣ on [-1,1]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

Key Concepts

Riemann Sums

Left, Right, and Midpoint Riemann Sums

Net Area

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