Approximating areas Estimate the area of the ...

Question

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.

(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says to use a left Riemann sum with equal to 4 subintervals to estimate the area under F of X, which is equal to 2 X2 + 1, from X equal to 0 to X equal to 2, and we're given 4 possible choices as our answers. For choice A, we have 7 divided by 2. For choice B, we have 13 divided by 2. For choice C, we have 11 divided by 2, and for choice D, we have 15 divided by 2. Now as to estimate the area underneath our curve using a left remo sum. So the first thing we want to do is determine our subinterval width, which we denote as delta X. I recall that delta X is going to be equal to our upper bound of our interval, which in our case is 2, minus the lower bound, which is 0 from the problem, and then that quantity is divided by N. So we're going to divide this by 4 since N is equal to 4 in our problem. And so we simplify this this expression, we see that delta X is going to be equal to 1 divided by 2. Next, we need to find the left end points for each of our sub-intervals. So for the first sub-interval, it's left endpoint, which will denote as X knot, is just going to be equal to the lower bound of our interval, which is going to be zero. And to find the next left end point for the next subinterval, which will denote as X1, this is just going to be equal to the previous left end point plus delta X. So our previous left endpoint is going to be 0. Then we'll have to add to that delta X, which is 1 divided by 2. And when we evaluate this expression, we see that X1 is going to be equal to 1 divided by 2. The next left endpoint for the next subinterval, which we'll denote as X2, is just going to be equal to our previous left endpoint, which is 1 divided by 2, plus delta X, which is 1 divided by 2, and so this is going to be equal to 1. And the endpoint for the last subinval, which will denote as X3. That is going to be equal to our previous left end point, which is going to be 1, plus, Delta X, which is 1 divided by 2. And so when we evaluate this expression, we see that X3 is going to be equal to 3 divided by 2. So now we have all the left end points, we can calculate our remod sum. So we call that our remod sum is going to be the sum. Of K equal to 02 3. Of F of X K multiplied by delta X. So this is going to be equal to the quantity of F of X0. Plus F of X1. Plus F of X2. Plus F of X3. In quantity multiplied by delta X. So we need to evaluate our function F of X at each of our left in points. So doing so for the first left endpoint, we'll have F of X, which is going to be equal to F of 0, and this is going to be equal to 2 multiplied by 0 squared plus 1, and this is just going to be equal to 1. For the next endpoint, we'll have F of X1, and this is going to be equal to F of 1 divided by 2, and this is going to be equal to 2 multiplied by the quantity of 1 divided by 2 in quantity squared plus 1. Which one we evaluate this expression is going to be equal to 3 divided by 2. For the next endpoint, we'll have F of X2, and this is going to be equal to F of 1, which is going to be equal to 2 multiplied by 1 squared plus 1. Which one we evaluate this expression is equal to 3. And finally, the last endpoint we'll have F of X3, and this is going to be equal to F of 3 divided by 2. This is going to be equal to 2, multiplied by the quantity of 3 divided by 2 and quantity squared plus 1. And when we evaluate this expression, we see that this is going to be equal 2. 11 divided by 2. So now that we have all those values, we can substitute them into our expression for our left red sum. In doing so, we see that our left 3 month sum is going to be equal to the quantity of 1 plus 3 divided by 2, plus 3 + 11 divided by 2 in quantity, this gets multiplied by delta X, which is 1 divided by 2. And when we evaluate this expression, we see that our left rement sum is going to be equal to 11 divided by 2. And so that is the answer that we are looking for for this problem. And if we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.
(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.

Key Concepts

Riemann Sums

Partitioning the Interval

Function Evaluation at Endpoints

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