{Use of Tech} Approximating net area The following functions a...

Question

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

ƒ(x) = 4 - 2x on [0,4]

(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

Hi everyone, let's take a look at this practice problem. This problem says let F of X be equal to 5 minus X on the closed interval from 2 to 6. Using N equal to 4 was the right remand sum approximation for the net area bounded by the graph of F and the X-axis on this interval. So this problem is asking us to calculate the right remand sum. 4 F of X on the closed interval from 2 to 6 using N equal to 4 subintervals. So the first thing we want to do is figure out the width of our intervals. And so for that, we'll be calculating delta X, and recall that this is going to be equal to the quantity of B minus A and quantity divided by N. For B here is the upper bound of our interval, A is the lower bound, and N is the number of subintervals. So using the information given twice in the problem, this is going to be equal to for our upper bound or interval, that's 6, and we'll have minus the lower bound of our interval, which is 2, and then that quantity is divided by the number of subintervals, which we're given as 4 in the problem. And when we evaluate this expression, we see that delta X is equal to 1. So now we can write down our 4 sub-intervals. So for our first sub-interval, we'll have the interval from 2 to 3. For the next subinterval we'll have the interval from 3 to 4. For the next interval, we'll have from 4 to 5, and for our last interval we'll have from 5 to 6. And since we're calculating the right remand sum, we want to use the right endpoint from all of these intervals. So, the right endpoint for our first Interval, which we'll call X1, that is going to be equal to 3. The right end point for a second interval, which we'll call X2, is equal to 4. The right endpoint of our third info, which we'll call X3, is equal to 5. And the right endpoint of our 4th inval, which we'll call X4, is equal to 6. So now we've identified the end points that we're going to use, let's try to calculate our remand sum. So recall that our remand sum is going to be the sum. From K equal to 1 to 4. Of F of X K. Multiplied by Delta X. And so this is going to be equal to the quantity of F of X1. Plus F of X2. Plus F of X3. Plus F of X4. In quantity multiplied by Delta X. So what we need to do is to evaluate our function F of X at each of our 4 endpoints. So for F of X1. This is going to be equal to F of 3. Which using the function that we're given to the problem, this is going to be equal to 5 minus 3, which is equal to 2. For the next end point, we'll have F of X2. Which is going to be equal to F of 4, which is equal to 5 minus 4, which is equal to 1. For the next interval, we'll have F of X3. Which is equal to F of 5. Which is equal to 5 minus 5, which is equal to 0. And for a final endpoint, we'll have F of X4. Which is going to be equal to F of 6. Which is equal to 5 minus 6, which is equal to -1. So substituting our values into our expression for the remand sum, we see that the right remand sum is going to be equal to. The quantity of 2 + 1 + 0 + minus 1. In quantity, multiplied by 1. And when we evaluate this expression, this is going to be equal to 2. So this here is the answer that we were looking for for this problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

ƒ(x) = 4 - 2x on [0,4]

(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 Sums

Net Area

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