Question

express the limit \\( \lim _{n \
ightarrow \infty} \sum_{i=1}^{n}\left(3\left(x_{i}^{*}\
ight)^{2}-6\left(x_{i}^{*}\
ight)^{5}\
ight) \delta x_{i} \\) over \\( 5,8 \\) as an integral.

provide \\( a, b \\) and \\( f(x) \\) in the expression \\( \int_{a}^{b} f(x) d x \\).

\\( a= \\) \\( b= \\) \\( f(x)= \\)

Explanation:

Step1: Recall the definition of definite integral

The definite integral \(\int_{a}^{b}f(x)dx=\lim_{n
ightarrow\infty}\sum_{i = 1}^{n}f(x_{i}^*)\Delta x_{i}\), where \([a,b]\) is the interval of integration and \(f(x)\) is the function.

Step2: Identify \(a\), \(b\) and \(f(x)\)

Given the interval \([5,8]\), by the definition of the definite integral, \(a = 5\) and \(b=8\).
Since \(\sum_{i = 1}^{n}(3(x_{i}^)^{2}-6(x_{i}^)^{5})\Delta x_{i}\), comparing with \(\sum_{i = 1}^{n}f(x_{i}^*)\Delta x_{i}\), we have \(f(x)=3x^{2}-6x^{5}\).

Answer:

\(a = 5\), \(b = 8\), \(f(x)=3x^{2}-6x^{5}\)