Question

exercises 1.2 the definite integral
score: 20/170 answered: 2/17
question 3
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express the limit \\(\lim_{n\to\infty} \sum_{i=1}^{n} \left(2(x_i^)^4 - 5(x_i^)^5\
ight) \delta x_i\\) over 4, 5 as an integral.
provide \\(a\\), \\(b\\) and \\(f(x)\\) in the expression \\(\int_{a}^{b} f(x) dx\\).
\\(a = \\), \\(b = \\), \\(f(x) = \\)
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Explanation:

Step1: Recall definite integral definition

The definite integral of $f(x)$ over $[a,b]$ is $\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x_i$, where $x_i^*$ are sample points in subintervals of $[a,b]$.

Step2: Identify $a, b, f(x)$

The interval is given as $[4,5]$, so $a=4$, $b=5$. Match the summand to $f(x_i^*)$: $f(x_i^*) = 2(x_i^*)^4 - 5(x_i^*)^5$, so $f(x)=2x^4 -5x^5$.

Answer:

$a = 4$, $b = 5$, $f(x) = 2x^4 - 5x^5$