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anonymous

  • 5 years ago

consider the integral \[\int\limits_{0}^{8} (x+e^x)^2 dx \] a. use the midpoint rule with n=4 to approximate this integral.

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  1. anonymous
    • 5 years ago
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    HELP! :(

  2. anonymous
    • 5 years ago
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    Expand the square, to get x^2+e^2x+2xe^x and integrate separately

  3. anonymous
    • 5 years ago
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    not really helping....

  4. anonymous
    • 5 years ago
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    i have to use the midpoint rule not just only integrate

  5. anonymous
    • 5 years ago
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    yes, so with the midpoint rule, you are asked to divide the curve into four rectangles. which means that \[\Delta = (b-a)/n \] where b = 8 and a = 0 and n =4 so the width of the sub intervals is \[\Delta=(8-0)/4 =2 \]

  6. anonymous
    • 5 years ago
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    \[\Delta x=\frac {b-a}{n}\rightarrow \frac {8-0}{4}=2\] your intervals are:\[[0,2],[2,4],[4,6],[6,8]\]

  7. anonymous
    • 5 years ago
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    So your subintervals are (0.2), (2,4),(4,6) and (6,8) which means your mid points are at x =1,3,5,7

  8. anonymous
    • 5 years ago
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    so evaluate at the midpoints for each of the subintervals: \[x=1,3,5,7\]

  9. anonymous
    • 5 years ago
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    thanks nadeem! big help :D

  10. anonymous
    • 5 years ago
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    therefore according to the rectangle rule, \[\int\limits_{0}^{8}f(x)dx \approx 2 * \sum_{1}^{7}f(x)\] where x takes values 1,3,5,7

  11. anonymous
    • 5 years ago
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    sure thing

  12. anonymous
    • 5 years ago
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    \[\int\limits_{0}^{1} \ln(x) / x ^{1/2} dx \] to approximate this integral using the trapezoid rule. can you also help me with this :( sorryy

  13. anonymous
    • 5 years ago
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    \[\frac{\Delta x}{2}=\frac{\frac{b-a}{n}}{2}\] how many subintervals do you need? you didn't specify Here is the generic equation: \[\int\limits\limits_{0}^{1} \frac{lnx}{\sqrt{x}}dx=\frac {\Delta x}{2}[\frac{lnx_0}{\sqrt{x_0}}+2*\frac {lnx_1}{\sqrt{x_1}}+...+2*\frac{lnx_{n-1}}{\sqrt{x_{n-1}}}+\frac {lnx_n}{\sqrt{x_n}}]\]

  14. anonymous
    • 5 years ago
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    the question is determine if the improper integral is convergent or divergent. if its convergent, find its value.

  15. anonymous
    • 5 years ago
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    Here is the problem..... you did posted earlier

  16. anonymous
    • 5 years ago
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    yeah sorry and i got this already thanks, but can you help me with other one i posted on the previous section?

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