{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.

f(x) = sin 2x on [0,3π/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 cosine of 3 X on the closed interval from 0 to pi divided by 2. Using N equal to 4, what is the left ribosome approximation for the net area bounded by the graph of F and the X-axis on this interval? And we're given 4 possible choices as our answers. For choice A, we have minus 1.146. For choice B, we have minus 0.098. For choice C, we have 0.197, and for choice D, we have 1.896. So for this problem, we need to calculate the left Riemann sum for our function F of X on the interval, on the closed interval from 0 to pi divided by 2. So the first thing we need to do is to determine our sub-interval width with N equal to 4 sub-intervals. And so that means that our sub-interval width delta X is going to be equal 2, and here we'll have the upper bound of our interval, which is going to be high divided by 2. Minus the lower bound of our interval, which is going to be 0, and then that quantity is going to be divided by N, which is 4. So, when we simplify this expression, we see that delta X is going to be equal to pi divided by 8. Next, we need to determine the left end points for each of our 4 sub-intervals. So, the left endpoint of our first subinterval, which is going to be called X1, is just going to be equal to the lower bound of our interval that we're looking at, which is going to be equal to 0. Did the left end point for the next sub interval, which we'll call X2, that's just going to be equal to X1 plus delta X. So we'll have 0 plus pi divided by 8, so we'll have pi divided by 8 as the second left endpoint. And for the remaining points, we'll just be adding delta X to the previous value. So that means that X3 is going to be equal to. Pi divided by 4, and X4 is going to be equal to 3 pi divided by 8. So now we can calculate our left free month sum, and so we call for our left free month sum, we're gonna have the sum of K equal to 1 to 4 of F of X, so 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 X of 3. Plus F of X 4. In quantity, multiplied by Delta X. So, in order to evaluate this remote sub, we need to calculate the value of our function F of X at each of our left end points. So, looking at the first end point, we'll have F of X1. And this is going to be equal to F of 0. Which is equal to. Cosine of the quantity of 3 multiplied by 0 in quantity. And when we evaluate this expression, we see that this is going to be equal to 1. Or the second endpoint will have F of X2, and that's going to be equal to F of pi divided by 8. Which is going to be equal to cosine of the quantity of 3 multiplied by high divided by 8. In quantity, and when we evaluate this expression, we see that it's equal to. 0.3827. For our next endpoint, we'll have F of X3, and that's going to be equal to F of high divided by 4. Which is going to be equal to cosine of quantity of 3 multiplied by pi divided by 4. In quantity, and when evaluating this expression, we see that it's equal to. Minus 0.7071. And for a final endpoint, we'll have F of X4. And this is going to be equal to F of 3 pi divided by 8. And this is going to be equal to cosine of the quantity of 3 multiplied by. 3 pi divided by 8. In quantity, evaluating this expression, we see that's equal to. Minus 0.9239. So, substituting these values into our expression for the Riemann sum, we see that a remand sum is going to be equal to. The quantity of one. Plus 0.38. 27 Plus -0.7071. Plus -0.92. 39. In quantity, and that whole quantity is going to be multiplied by Delta X, which is pi divided by 8. And so when we evaluate this expression, we see that it is going to be equal to. Minus 0.098. So here we just round it to three decimal places, and this here is the answer that we're looking for for this problem. And 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.

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

f(x) = sin 2x on [0,3π/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

Definite Integral

Trigonometric Functions

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