(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 sum for the function f(x) = x + 3, given an arbitrary partition P = {X0, X1, X2, ..., Xn} of the interval [a, b], you can follow these steps:
Step 1: Calculate the subinterval lengths
To construct the lower and upper sums, you need to determine the lengths of each subinterval. The length of a subinterval is given by Δxi = Xi - Xi-1, where Xi and Xi-1 are consecutive partition points.
Step 2: Calculate the infimum and supremum in each subinterval
For each subinterval [Xi-1, Xi], you need to find the infimum and supremum of f(x) within that interval. The infimum (denoted by m) is the smallest value that f(x) takes in the interval, while the supremum (denoted by M) is the largest value. Since f(x) = x + 3 is an increasing function, the infimum and supremum coincide with the function value at each endpoint of the subinterval.
Step 3: Calculate the lower sum
The lower sum (L) is obtained by summing up the products of the infimum and subinterval lengths for each subinterval. Mathematically, L = Σ(m * Δxi), where the summation is taken over all subintervals.
Step 4: Calculate the upper sum
The upper sum (U) is obtained in a similar way by summing up the products of the supremum and subinterval lengths for each subinterval. Mathematically, U = Σ(M * Δxi), where the summation is taken over all subintervals.
Now, let's move on to part (B) and use the answers from part (A) to evaluate the integral of f(x) from a to b.
Step 5: Evaluate the integral
The integral of f(x) from a to b, denoted by ∫[a,b] f(x) dx, is defined as the limit of the lower sums as the maximum subinterval length approaches zero. Similarly, it is also equal to the limit of the upper sums.
To evaluate the integral using the lower sum, compute L as described in step 3. To evaluate the integral using the upper sum, compute U as described in step 4. Then, compare the values of L and U at the end of the computations.
Both the lower sum and upper sum will converge to the same value as the maximum subinterval length approaches zero. This value represents the exact integral of f(x) from a to b.