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Express: \lim_{n\rightarrow \infty} \sum_{i=1}^{n} 17x^9_i \Delta x as a definite integral for...

Question:

Express: {eq}\lim_{n\rightarrow \infty} \sum_{i=1}^{n} 17x^9_i \Delta x {/eq} as a definite integral for {eq}a=-4, \ b=4 {/eq} and evaluate the definite integral.

Summation of Limit:


Suppose we have an integral expression {eq}\displaystyle \int_{a}^{b}f(x)\ dx {/eq} and the values of the lower and upper boundaries of the definite integral are given, then we'll compute the expression for the function {eq}f(x) {/eq} by the general formula that is shown below:

{eq}\displaystyle \lim_{n\rightarrow \infty} \sum_{i=1}^{n} f(x_i)\Delta x=\int_{a}^{b}f(x)\ dx {/eq}

After that, we'll use the general property of definite integral to solve the obtained expression.

{eq}\displaystyle\int_{-a}^{a}f(x)\ dx=0, \textrm{ if f(x) is odd}\\ \displaystyle\int_{-a}^{a}f(x)\ dx=2\int_{0}^{a}f(x)\ dx, \textrm{ if f(x) is even}\\ {/eq}

Answer and Explanation:


The given summation notation with limit at infinity is:

{eq}\displaystyle \lim_{n\rightarrow \infty} \sum_{i=1}^{n} 17x_i^9 \Delta x=? {/eq}

The lower and upper boundaries of the definite integral is:

{eq}a=-4 \\ b=4 {/eq}

By the general summation notation {eq}\displaystyle \lim_{n\rightarrow \infty} \sum_{i=1}^{n} f(x_i)\Delta x {/eq} and the given summation notation, we have:

{eq}f(x_i)=17x_i^9 {/eq}

Replacing the variable {eq}x_i {/eq} by the variable x in the above expression, we get:

{eq}f(x)=17x^9 {/eq}


Now, the expression for the definite integral expression is:

{eq}\begin{align*} \displaystyle\int_{a}^{b}f(x)\ dx&=\int_{-4}^{4}17x^9\ dx\\ &=17\int_{-4}^{4}x^9\ dx\\ \end{align*} {/eq}

We can see that the expression {eq}x^9 {/eq} is an odd function as we get {eq}(-x)^9=-x^9\rightarrow f(-x) =-f(x) {/eq}.

Thus, the value of the definite integral expression by the integral property is:

{eq}\begin{align*} \displaystyle17\int_{-4}^{4}x^9\ dx&=17(0)\\ &=\boxed0 \end{align*} {/eq}


Learn more about this topic:

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Evaluating Definite Integrals Using the Fundamental Theorem

from AP Calculus AB: Exam Prep

Chapter 16 / Lesson 2
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