{Use of Tech} Midpoint Riemann sums with a calculator

Question

{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.

(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₁⁴ 2√𝓍 d𝓍

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to estimate the integral from 1 to 6 of the quantity of X2 + 1 in quantity DX with the midpoint Riemann sum or N equal to 20, in equal to 50, and N equal to 100 subintervals. So our first step in solving the problem is to determine our interval width. And so recall that our interval width delta X. is equal to the quantity of B minus a in quantity divided by n, where B here is the upper limit of our integration, A is our lower limit, and N is the number of some intervals. We'll have to calculate this 3 different times for each of our n values. So 4 N equal to 20. We'll have that delta X is going to be equal to for our upper bound of our integral, we had 6, we'll have minus our lower bound, which is 1, and then that quantity is going to be divided by 20. And when we evaluate this expression, we see that this is equal to 0.25. We'll do something similar, 4 and equal to 50. And calculating delta X, this is going to be equal to the quantity of 6 minus 1 in quantity, divided by 50, which we, when we evaluate this expression, this is equal to 0.1. Then finally, 4 N equal to 100. We'll have the delta X equal to the quantity of 6 minus 1 in quantity, divided by 100, which is going to be equal to 0.05. So now that we have the delta X, we just need to calculate the midpoint remand sum, and recall that the midpoint remand sum, which in this case, we'll label as MN, is going to be equal to the sum. Of I equal to 1 ton. Of the quantity, and here we'll have to use our integram, so we have X2 + 1. Now, for X2, we're going to write that as the quantity of 1 plus the quantity of I minus 12 in quantity, multiplied by delta X. In quantity and then I get squared since our X value is being squared. And then we'll have plus one and quantity, and that is all multiplied by Delta X. So, we're going to have to calculate this midpoint remod sum, or the 3 different end values. So for N equal to 20, We'll have that M 20 is going to be equal to the sum. Of I equal to 1 to 20. Of the quantity of the quantity of 1, plus the quantity of I minus 1/2 in quantity, multiplied by delta X, which in this case is 0.25. In quantity where Plus one In quantity, multiplied by 0.25. Now, we can actually evaluate the sum using our calculators, and in doing so, we see that this is going to be approximately equal to 76.64. We'll do something similar for N equal to 50, so we'll have M 50 for the remod sum, and this is going to be equal to the sum of I equal to 1 to 50 of the quantity of the quantity of 1 plus the quantity of I minus 12 in quantity multiplied by delta X, which in this case is 0.1. In quantity where. Plus 1 in quantity multiplied by 0.1. And when we evaluate this expression using our calculator, we see that it's approximately equal to 76.66. And the final remand sum that we need to evaluate is going to be M 100, and this is going to be equal to the sum of I equal to 1 to 100 of the quantity of the quantity of 1 plus the quantity of I minus 12. In quantity multiplied by Delta X, which is 0.05. In quantity squared. Plus one. In quantity multiplied by 0.05. And evaluating this expression using our calculators, we'll see that this is approximately equal to. 76.666. And so what we see is that as we increase the end value, we're actually approaching 76 and 2/3, which can be rewritten as 230 divided by 3. So that means that we can estimate that the integral from 1/6 of the quantity of X2 plus 1 in quantity DX is going to be equal to. 230 Divided by 3. So that is the answer that we're 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} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₁⁴ 2√𝓍 d𝓍

Key Concepts

Definite Integral

Riemann Sums

Midpoint Rule

Watch next