{Use of Tech} Approximating definite integrals with a ...

Question

{Use of Tech} Approximating definite integrals 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.

∫₀¹ (𝓍² + 1) d𝓍

Accepted Answer

Welcome back, everyone. In this problem, using a right endpoint Riemann sum, find the exact value of the definite integral between 2 and 4 of 3 X squared with respect to X. A says it's 72, B 56, C 40, and the D 24. If we're going to find the right endpoint Raman sum, it would be good for us to first find the width of our sub interval. So now here. We know that the width of our subinterval is going to be equal to the difference between the upper and lower limits that's 4 minus 2 divided by n, which is equal to 2 divided by n. No, to find the red and point, OK. Right end point. It's going to be equal to 2 + I multiplied by our width of 4 subinterval. So that's 2 + 2 I divided by N. So now when we plug this into our function, this value of our right endpoint Xi is going to be equal to 3 multiplied by X or 2 + 21 divided by n because we substitute that for X squared, which is going to be 3 multiplied by 4 + 8 I divided by N + 4 I 2 divided by N2 when we expand it. And that is going to be 12 + 24 I divided by N plus 12 +12 divided by N2 when we distribute the 3. So no, this implies that the Riemann sum on the right in abstract terms is going to be equal to the sum from 1 ton in sigma notation of 12 + 24 I divided by N plus 12 I square divided by n2 multiplied by 2 divided by n. And now when we expand that 2 divided by N term, then we get the sum from I ton of 24 divided by N plus 48 I divided by N squared. Plus 24 I squared divided by n cubed. Now where do we go from here? Well, using the sum from I ton of 1 to be equal to n. OK. The sum from I to end, OK. From 1 ton of i to be equal to n multiplied by n + 1 divided by 2, OK, and the sum from I ton of i squared to be equal to n multiplied by n + 1 multiplied by 2 n + 1 divided by 6. OK then. We can say that for a large n. N + 1 divided by N will be 10 to 1, sorry. And N + 1, OK, multiplied by 2 N + 1. OK. Is going divided by n2 is going to 10 to 2. Because if we were to substitute these values of, of 1 I and I squared into our Riemann sum, OK, then these terms will be there and eventually they'll cancel out. In other words, RN would be equal to 24 plus 24 multiplied by N + 1, OK, divided by N. Plus 4 multiplied by n + 1 multiplied by 2 n + 1 divided by 6 n squared. And that, based on what we know they're going to tend to, will be equal to 24 plus 24 multiplied by 1 plus 4 multiplied by 2. That's going to be 24 + 24 + 8, and that is equal to 56. Does the write the exact value? Uh, for integral between 2 to 4 of 3 X squared with respect to X is going to be equal to 56. And that means as N increases, the sum approaches 56. Therefore, B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

{Use of Tech} Approximating definite integrals 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.

∫₀¹ (𝓍² + 1) d𝓍

Key Concepts

Definite Integral

Riemann Sum

Numerical Integration

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