If I have y=e^x and know it represents the left Riemann sum with n=2 approximating ∫ with upper limit of 1 and lower limit of 0 e^x dx. How do I write out the terms of the sum without evaluating it? Similarly, how would I do that with the right Riemann sum?
clearly you need to review Riemann sums. This would be
left sum = 1/2 (f(0) + f(1/2))
right sum = 1/2 (f(1/2) + f(1))
Right, but I have to only put down two things that get added together. I do not understand how to include the 1/2. Without the 1/2, wouldn't it be 1+e^(1/2) for the left and then e^(1/2)+e for the right? But then how can I show the 1/2 when only able to show two things being added?
Really? If you only want the sum, then just distribute the 1/2 over each term
e(0)/2 + e(1/2)/2
e(1/2)/2 + e/2
To understand how to write out the terms of the left Riemann sum and the right Riemann sum without evaluating them, we need to start by understanding the general formulas for these sums.
The left Riemann sum with n subintervals is given by the formula:
L_n = ∑ f(x_i) * Δx,
where ∑ represents the sum, f(x_i) is the value of the function at each left endpoint of the subinterval, Δx is the width of each subinterval, and i is the index representing the subintervals.
Similarly, the right Riemann sum with n subintervals is given by the formula:
R_n = ∑ f(x_i+1) * Δx,
where f(x_i+1) is the value of the function at each right endpoint of the subinterval.
Now, let's apply these formulas to your specific problem, where y = e^x and we have n = 2 subintervals approximating ∫[0,1] e^x dx.
For the left Riemann sum:
1. Divide the interval [0, 1] into n = 2 equal subintervals.
- The left endpoints of the subintervals will be: x_0 = 0 and x_1 = 1/2.
- The width of each subinterval (Δx) will be: Δx = (1 - 0)/2 = 1/2.
2. Evaluate the function e^x at the left endpoints of each subinterval.
- At x_0 = 0: f(x_0) = e^0 = 1.
- At x_1 = 1/2: f(x_1) = e^(1/2).
3. Write out the terms of the sum without evaluating it:
L_2 = f(x_0) * Δx + f(x_1) * Δx
= 1 * (1/2) + e^(1/2) * (1/2)
= 1/2 + (e^(1/2))/2
For the right Riemann sum:
1. Divide the interval [0, 1] into n = 2 equal subintervals.
- The right endpoints of the subintervals will be: x_1 = 1/2 and x_2 = 1.
- The width of each subinterval (Δx) will be: Δx = (1 - 0)/2 = 1/2.
2. Evaluate the function e^x at the right endpoints of each subinterval.
- At x_1 = 1/2: f(x_1) = e^(1/2).
- At x_2 = 1: f(x_2) = e^1 = e.
3. Write out the terms of the sum without evaluating it:
R_2 = f(x_1) * Δx + f(x_2) * Δx
= e^(1/2) * (1/2) + e * (1/2)
= (e^(1/2))/2 + e/2
Note that while these expressions are not evaluated, they represent the respective left and right Riemann sums for approximating the integral of e^x on the interval [0, 1] with two subintervals.