{Use of Tech} Approximating definite integrals

Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.

(c) Calculate the left and right Riemann sums for the given value of n.

∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to calculate the left and right remand sums for the interval from 0 toi of sin of X DX using N equal to 4 sub-intervals. So, our first step in calculating our remand sums is to determine the width of our subintervals. And so that's going to be Delta X, and recall that delta X is going to be equal to. B minus A, and that quantity is going to be divided by n, where B here is the upper bound of our interval, A is the lower bound of our integral, and N is the number of subintervals. So substituting in the values that we have, this is going to be equal to the quantity of pi minus 0. In quantity, divided by 4. So, Delta X is going to be equal to pi divided by 4. Next, we need to figure out the endpoints of our sub-intervals. So, for the first interval, we'll have X not being equal to 0. X1 is going to be equal to pi divided by 4. For the next interval, its end points are going to be X1 and X2, which is going to be equal to pi divided by 2. For the next interval we'll have X2, and X3, which is going to be equal to 3 pi divided by 4 for its endpoints, and for the final interval we'll have X3 as an endpoint and X4, which is going to be equal to pi. So now we can calculate our remand sum. So, we'll start with the left remand sum. So we'll call this L4, since we have 4 subintervals, and this is going to be equal to the quantity, and here for sin of X, we'll substitute for X, each of the left end points for our subintervals. So we'll have the quantity of sine of X0. Plus the sign of X1. Plus the sign of X2. Plus the sign of x3. And then that quantity is going to be multiplied by delta X. So, substituting in our values for X and delta X, we see that this is going to be equal to the quantity, the sign of 0, plus the sign of pi divided by 4. Plus the sign of pi divided by 2. Plus the sign of 3 pi divided by 4. In quantity multiplied by pi divided by 4. So evaluating this expression, this is going to be equal to the quantity of 0, plus the square root of 2 divided by 2, plus 1 plus the square root of 2 divided by 2 in quantity multiplied by pi divided by 4, and simplifying this expression, this is going to be equal to pi divided by 4 multiplied by the quantity of 1 plus the square root of 2 in quantity. And so that is the value that we're looking for for the left Riemann sum. So now we just need to calculate our right remand sum, which we'll label as R4. And this is going to be equal to the quantity, and here for sin of X. We'll be substituting in for X, the right in points for each of our intervals. So we'll have the quantity of sine of X1. Plus, Sign of X2. Plus sign of X3. Plus sine of x 4. In quantity multiplied that by delta X. So substituting in our values for X and delta X, we see that this is going to be equal to the quantity of the sign of pi divided by 4. Plus the sign of pi divided by 2. Plus sign. Of 3 pi divided by 4. Plus the sign of pie. And that quantity is multiplied by pi divided by 4. So, evaluating this expression, this is going to be equal to the quantity of the square root of 2, divided by 2. Plus 1 plus the square root of 2 divided by 2, plus 0 in quantity multiplied by pi divided by 4. Which simplify this expression is going to be equal to pi divided by 4, multiplied by the quantity of 1, plus the square root of 2 in quantity. And so that's the answer that we're looking for for the right Riemann sum. And notice that both our left and right Riemann sums are equal to each other. So we can just rewrite this as our final answer is L4 equal to R4, which is equal to pi divided by 4, multiplied by the quantity of 1 plus the square root of 2 in quantity. So that is the final answer that we are 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} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.

∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

Key Concepts

Riemann Sums

Definite Integral

Subintervals and n

Watch next