Final answer:
To calculate the lower sum and upper sum of a function defined on a given interval and partition, evaluate the function at each subinterval's left and right endpoints respectively, multiply it by the subinterval's length, and sum the products. For the function f(x) = x³ on the interval [0, 1] and the partition P = {0, 0.1, 0.4, 1}, the lower sum L(P, f) and upper sum U(P, f) are the same due to the function being strictly increasing.
Step-by-step explanation:
The function f(x) is defined as f(x) = x³ on the interval [0, 1]. The partition P is given as {0, 0.1, 0.4, 1}. To compute the lower sum L(P, f), we evaluate f(x) at each subinterval's left endpoint and multiply it by the length of the subinterval. The lower sum is the sum of these products. Similarly, to compute the upper sum U(P, f), we evaluate f(x) at each subinterval's right endpoint and multiply it by the length of the subinterval. The upper sum is the sum of these products.
For the given partition P and function f(x) = x³, the lower sum L(P, f) and upper sum U(P, f) are:
L(P, f) = [0.1³ * (0.1 - 0) + 0.4³ * (0.4 - 0.1) + 1³ * (1 - 0.4)]
U(P, f) = [0.1³ * (0.1 - 0) + 0.4³ * (0.4 - 0.1) + 1³ * (1 - 0.4)]