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Numerical Integration In Exercises $81-84,$ use the Trapezoidal and Simpsons to approximate the value of the definite integral. Let $n=4$ and round your answer to four decimal places. Use a graphing utility to verify your result. $$ \int_{0}^{4} \frac{8 x}{x^{2}+4} d x $$

Numerical Integration In Exercises $81-84,$ use the Trapezoidal and Simpsons to approximate the value of the definite integral. Let $n=4$ and round your answer to four decimal places. Use a graphing utility to verify your result. $$ \int_{0}^{4} \frac{8 x}{x^{2}+4} d x $$

Calculus of a Single Variable

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Carson Merrill verified

Numerade educator

Determine a definite integral that represents the arc length of ( r=4+9 sin ( heta), 0 leq heta leq 2 pi ). [ int_{0}^{2 pi} ] ( d heta )

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Sydney Bement verified

Numerade educator

c) What can we say about the original series, ( sum_{n=2}^{infty} frac{1}{sqrt{4 n-5}} ) ? It converges, because the integral converges, It converges, because the integral diverges. It diverges, because the integral diverges. It diverges, because the integral converges. Since the integral doesn't converge or diverge, we cannot determine if this series converges.

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Khushbu Rani verified

Numerade educator

If you were to solve this using integration by parts, what should you choose for your u and dv ? ? x^6 arctan(9x) dx u = dv =

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Zhumagali Shomanov verified

Numerade educator

Find all the values of x for which the given power series converges. Use interval notation with exact values. ?_{n=1}^{?} (7x)? / n¹? The series is convergent for x in the interval You must show your work in order to get credit for this. Be sure to clearly state what test(s) you are using to determine convergence.

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William Semus verified

Numerade educator

Here is a graph of r = 5 sin(4?) Determine a definite integral that represents the area of the region enclosed by one petal of the function. ??? [ ] d? Bounds: a = 0 to b =

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Zhumagali Shomanov verified

Numerade educator

Consider the series ?_{n=2}^{?} rac{1}{?{4n-5}} a) Explain why we cannot use the Divergence Test here. b) We can use the Integral Test. Does this related integral converge or diverge? ?_{2}^{?} rac{dx}{?{4x-5}} The integral converges. The integral diverges. The integral neither converges nor diverges. Show work to justify your answer.

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\begin{tabular}{|r|r|r|r|} \hline\( x \) & \( f(x) \) & \( f^{\prime}(x) \) & \( f^{\prime \prime}(x) \) \\ \hline 8 & 2 & 15 & \( -10 \) \\ \hline 13 & 12 & 27 & \( -19 \) \\ \hline 18 & 7 & 25 & \( -16 \) \\ \hline 23 & 4 & 26 & \( -12 \) \\ \hline 28 & 14 & 30 & \( -4 \) \\ \hline 33 & 9 & 34 & \( -9 \) \\ \hline 38 & 13 & 3 & \( -7 \) \\ \hline \end{tabular} a) Use Simpson's Rule and the data in the above table to estimate the value of the integral \( \int_{8}^{38} f(x) d x \) as accurately as possible. (You may not need all of the data.) \[ \int_{8}^{38} f(x) d x \approx \] b) Assume we are interested in the Taylor Series for \( f(x) \), centered at \( x=8 \). What is \( P_{2} \), the Taylor Polynomial of degree two, also centered at 8 ? \[ P_{2}= \] \( > \) Next Question

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