(A) Given that P = {X0, X1, X2.......Xn} is an arbitrary partition of [a,b], find the lower and upper sum for f(x)=x+3.
(B) Use your answers to part (a) to evaluate the integral of f(x) from a to b.
To find the lower and upper sums for the function f(x) = x + 3, given an arbitrary partition P = {X0, X1, X2, ..., Xn} of the interval [a, b], follow these steps:
1. Calculate the length of each subinterval, Δxi, in the partition. This is done by subtracting the right endpoint from the left endpoint for each subinterval: Δxi = Xi - Xi-1.
2. Calculate the lower sum, L(f, P), by adding up the product of the infimum of f(x) over each subinterval and the corresponding subinterval length. The infimum of f(x) is the smallest possible value of f(x) over each subinterval. In this case, the infimum of f(x) = x + 3 is achieved when x takes its smallest value in each subinterval, which is Xi-1.
L(f, P) = Σ[f(xi-1) * Δxi] for i = 1 to n, where xi-1 is the left endpoint of each subinterval and Δxi is the length of each subinterval.
3. Calculate the upper sum, U(f, P), by adding up the product of the supremum of f(x) over each subinterval and the corresponding subinterval length. The supremum of f(x) is the largest possible value of f(x) over each subinterval. In this case, the supremum of f(x) = x + 3 is achieved when x takes its largest value in each subinterval, which is Xi.
U(f, P) = Σ[f(xi) * Δxi] for i = 1 to n, where xi is the right endpoint of each subinterval and Δxi is the length of each subinterval.
Now let's calculate the lower and upper sums for f(x) = x + 3 using the given arbitrary partition P = {X0, X1, X2, ..., Xn}.
(A) Lower and Upper Sums:
Lower sum, L(f, P) = Σ[f(xi-1) * Δxi] for i = 1 to n
Upper sum, U(f, P) = Σ[f(xi) * Δxi] for i = 1 to n
(B) To evaluate the integral of f(x) from a to b using the lower and upper sums, take the limit of the lower and upper sums as the number of subintervals approaches infinity (when the maximum length of any subinterval approaches zero):
∫(from a to b) f(x) dx = lim[Σ[f(xi) * Δxi] - Σ[f(xi-1) * Δxi]] as n approaches infinity
I hope this helps you understand how to find the lower and upper sums for an arbitrary partition and how to evaluate the integral of a function using these sums.