Use the concept of the definite integral to find the total area between the graph of f(x) and the x-axiİs, by taking the limit of the associated right Riemann sum. Write theexact answer. Do not round. (Hint: Extra care is needed on those intervals wheref(x) <0. Remember that the definite integral represents a signed area.)f(x) = 5x + 2 on [0, 2]%3DAnswerO KeypadKeyboard ShortcutSubmit Answer© 2020 Hawkes LearningType here to search

Answered: Use the concept of the definite…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

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Chapter2: Second-order Linear Odes

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Use the concept of the definite integral to find the total area between the graph of f(x) and the x-axiİs, by taking the limit of the associated right Riemann sum. Write the
exact answer. Do not round. (Hint: Extra care is needed on those intervals wheref(x) <0. Remember that the definite integral represents a signed area.)
f(x) = 5x + 2 on [0, 2]
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Expert Solution

Step 1

Let $f (x)$ be continuous and non-negative function defined on the closed interval $[a, b]$ . Then the definite integral of the function $f (x)$ over the interval $[a, b]$ , denoted by $\int_{a}^{b} f (x) d x$ , is the limit of a Riemann sum as the norm of the partition approaches zero or number of subdivisions approaches infinity.

i.e. $\int_{a}^{b} f (x) d x = \lim_{n \to \infty} \sum_{i = 1}^{n} ∆ x_{i} f (x_{i}) = \lim_{n \to \infty} \sum_{i = 1}^{n} ∆ x \cdot f (x_{i})$ , where $∆ x = \frac{b - a}{n} a n d x_{i} = a + ∆ x \cdot i$

Step 2

The area under the graph $f (x) = 5 x + 2$ over the interval $[0, 2]$ is given by the shaded region in the figure

Here, $∆ x = \frac{2}{n} a n d x_{i} = (0 + \frac{2}{n} \cdot i) = \frac{2 i}{n}$

Then $\int_{0}^{2} (5 x + 2) d x = \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{2}{n} \cdot f (\frac{2 i}{n})$

$= \lim_{n \to \infty} \frac{2}{n} \sum_{i = 1}^{n} [5 (\frac{2 i}{n}) + 2] = \lim_{n \to \infty} \frac{4}{n} \sum_{i = 1}^{n} [\frac{5 i}{n} + 1]$

Thus $\int_{0}^{2} (5 x + 2) d x = = \lim_{n \to \infty} \frac{4}{n} \sum_{i = 1}^{n} [\frac{5 i}{n} + 1] - - - - - - (1)$

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