19-22. {Use of Tech} Trapezoid Rule approximations. Find the i...

Question

19-22. {Use of Tech} Trapezoid Rule approximations. Find the indicated Trapezoid Rule approximations to the following integrals.

21. ∫(0 to 1) sin(πx) dx using n = 6 subintervals

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to approximate the interval from 0 to 2 of cosine of the quantity of pi multiplied by X in quantity DX using the trapezoid rule with equal to 4 subintervals. And we're given 4 possible choices as our answers. For choice A, we have minus 2, for choice B, we have 0. Choice C, we have minus 1, and for choice D, we have 1. So we need to approximate this interval, using the trapezoid rule with N equal to 4 sub-intervals. So the first thing we want to do is determine our sub-interval width, which we'll call delta X. And recall that our subinterval width is going to be equal to, and here we'll take the upper bound of our limits of integration, which will be 2 minus our lower bound, which in this case is 0, and we'll take that quantity and divide it by n, which in our case is equal to 4. And so when we evaluate this expression, we see that delta X is going to be equal to 1 divided by 2. Next, we need to determine the X values of the endpoints for each of our sub-intervals. So, our first endpoint is going to be called X0, and this is just gonna be equal to the lower bound of our limits of integration, which in our case, is going to be equal to 0. And then for each successive endpoint, we'll need to add delta X up until we reach X equal to 2. So, our next endpoint is going to be X1, and this is just going to be equal to 0 + 1/2, which is going to give me 1/2. Then we'll have 4 X2, we'll have 1/2 plus 1/2, which is going to be equal to 1. For X3, when we add half, this is going to be equal to 3 divided by 2, and for X4, this is going to be equal to 2. When we add 1/2, 23 divided by 2. So now that we have our endpoints, we can look at using our trapezoid rule. So, recall that 4 trapezoid rule that we have the end rule from A to B of F of X. DX is going to be approximately equal to. Delta X divided by 2, multiplied by the quantity, and herefore in equal to 4, we're going to have F of X0. Plus 2 multiplied by F of X1. Plus 2 multiplied by F of X. 2. Plus 2 multiplied by F of X3. Plus F of X4. In quantity So, in order to evaluate this expression, we need to evaluate our function F at each of the endpoints, where F of X here is just going to be our integram, which is going to be the cosine of the quantity of pi X in quantity. So, we'll need to evaluate this function at each of our endpoints. So for the first endpoint, we'll have F of X0. Which is going to be equal to F of 0. Which is equal to cosine of the quantity of pi multiplied by zero in quantity, and when we evaluate this expression, we see that this is going to be equal to 1. The next end point will have F of X1. Which is going to be equal to F of 1 divided by 2. Which is equal to cosine. Of the quantity of pi, multiplied by 1 divided by 2 in quantity, which when we evaluate this expression is going to be equal to 0. For the next point in point we'll have F of X2. Which is going to be equal to F of 1. Which is equal to cosine of the quantity of pi multiplied by 1 in quantity, which when we evaluate this expression is going to be equal to -1. For the next end point, we'll have F of X3. And this is going to be equal to F of 3 divided by 2. Which is equal to cosine of the quantity of pi, multiplied by 3 divided by 2. In quantity, which when we evaluate this expression is equal to 0. And for the final endpoint, we have F of X4. Which is going to be equal to. F of 2, which is equal to cosine. Of the quantity of pi multiplied by 2 in quantity, which when we evaluate this expression is going to be equal to 1. So now we have all the information that we need to apply our trapezoid rule to our integral. So, the integral. From 0 to 2 of cosine of the quantity of pi multiplied by X and quantity DX is going to be approximately equal to. And here we'll have for delta X, we have 1 divided by 2, and then that quantity is going to be divided by 2 again. And then that gets multiplied by the quantity, and here we'll substitute in our values for the F of X, evaluated all the end points, so we'll have one. Plus 2 multiplied by 0, plus 2 multiplied by -1. Plus 2 multiplied by 0 + 1 in quantity. And when we evaluate this expression, we see that this is going to be equal to 0. So that is the answer that we were 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.

19-22. {Use of Tech} Trapezoid Rule approximations. Find the indicated Trapezoid Rule approximations to the following integrals.
21. ∫(0 to 1) sin(πx) dx using n = 6 subintervals

Key Concepts

Trapezoidal Rule

Subintervals

Definite Integral

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