Write the limit as a definite integral on the interval where is any point in the subinterval.

Question

Quizplus · Accepted Answer

The given limit can be rewritten as: \lim_{n o\infty}\sum_{i=1}^{n}f(x_i)\Delta x where f(x) = 11ec1854_fd9d_0b96_acfc_e3be7471efcb_TBX8702_11 and Delta x = (11ec1854_fd9d_0b99_acfc_5dbe92636a0f_TBX8702_11 - 11ec1854_fd9d_0b97_acfc_8f937dce9138_TBX8702_11)/n Let S_i = (11ec1854_fd9d_0b97_acfc_8f937dce9138_TBX8702_11 + iDelta x) be any point in the i-th subinterval [_,]. Then, the definite integral on the interval [_,] is given by: int_{_}^{}f(x)dx = lim_{n o\infty}\sum_{i=1}^{n}f(S_i)\Delta x Comparing this with the original expression for the limit, we see that it is the same as the limit as n goes to infinity of the Riemann sum for the definite integral of f(x) over the interval [_,] with S_i as the sample point for each subinterval. Therefore, the limit is equal to the definite integral of f(x) over the interval [_,]. Substituting the values given in the answer choices, we can see that the interval [11ec1854_fd9d_32ab_acfc_3ff9f5f11ceb_TBX8702_11, 11ec1854_fd9d_32ac_acfc_87fb911c0dd7_TBX8702_11] is the only one that lies within the interval [11ec1854_fd9d_0b97_acfc_8f937dce9138_TBX8702_11, 11ec1854_fd9d_0b99_acfc_5dbe92636a0f_TBX8702_11], and hence, is the correct choice.

Write the Limit as a Definite Integral on the Interval