consider the integral (4x^2+2x+4)dx from 0 to 6
Find the Riemann sum for this integral using right endpoints and n=3
and
Find the Riemann sum for this same integral, using left endpoints and n=3
To find the Riemann sum for the given integral using right endpoints and n = 3, we can follow these steps:
1. Calculate the width of each subinterval, Δx, by dividing the total interval length (6 - 0 = 6) by the number of subintervals (n = 3): Δx = (6 - 0) / 3 = 2.
2. Determine the right endpoints of each subinterval by adding Δx to the starting point of each subinterval. In this case, we start at x = 0, so the right endpoints are: x1 = 0 + 2 = 2, x2 = 2 + 2 = 4, x3 = 4 + 2 = 6.
3. Evaluate the function at each right endpoint to find the corresponding function values. For each subinterval, we evaluate the function f(x) = 4x^2 + 2x + 4 at the right endpoint. So, f(2) = 4(2)^2 + 2(2) + 4 = 24, f(4) = 4(4)^2 + 2(4) + 4 = 84, f(6) = 4(6)^2 + 2(6) + 4 = 196.
4. Multiply each function value by the width of the corresponding subinterval. In this case, the width of each subinterval is Δx = 2. So, the products are: 24 * 2 = 48, 84 * 2 = 168, 196 * 2 = 392.
5. Finally, add up all the products to find the Riemann sum: 48 + 168 + 392 = 608.
Therefore, the Riemann sum for the given integral using right endpoints and n = 3 is 608.
To find the Riemann sum for the same integral using left endpoints and n = 3, we can follow similar steps:
1. Calculate the width of each subinterval, Δx, as before: Δx = (6 - 0) / 3 = 2.
2. Determine the left endpoints of each subinterval by subtracting Δx from the starting point of each subinterval. In this case, we start at x = 0, so the left endpoints are: x0 = 0, x1 = 0 + 2 = 2, x2 = 2 + 2 = 4.
3. Evaluate the function at each left endpoint to find the corresponding function values. For each subinterval, we evaluate the function f(x) = 4x^2 + 2x + 4 at the left endpoint. So, f(0) = 4(0)^2 + 2(0) + 4 = 4, f(2) = 4(2)^2 + 2(2) + 4 = 24, f(4) = 4(4)^2 + 2(4) + 4 = 84.
4. Multiply each function value by the width of the corresponding subinterval. In this case, the width of each subinterval is Δx = 2. So, the products are: 4 * 2 = 8, 24 * 2 = 48, 84 * 2 = 168.
5. Finally, add up all the products to find the Riemann sum: 8 + 48 + 168 = 224.
Therefore, the Riemann sum for the given integral using left endpoints and n = 3 is 224.