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anonymous

  • one year ago

Write the Riemann sum to find the area under the graph of the function f(x) = x4 from x = 5 to x = 7.

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  1. anonymous
    • one year ago
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    https://i.gyazo.com/5703a9ba56f6d799fdd797a328f5c247.png

  2. jim_thompson5910
    • one year ago
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    |dw:1441763773280:dw| Hint: \[\Large \text{area under f(x) from a to b} = \lim_{n \to \infty} \sum_{i=1}^{n}f(x_i)*\Delta x\] where \[\Large \Delta x =\frac{b-a}{n}\] \[\Large x_i = a+i*\Delta x\]

  3. anonymous
    • one year ago
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    I went ahead and did this and got it wrong. Ended up with another one. So can you solve this one as an example @jim_thompson5910

  4. jim_thompson5910
    • one year ago
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    if a = 5 and b = 7, what is delta x equal to?

  5. jim_thompson5910
    • one year ago
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    any ideas?

  6. anonymous
    • one year ago
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    Wouldn't it equal 0? b is 7, a is 5, and n is infinity?

  7. anonymous
    • one year ago
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    @jim_thompson5910

  8. jim_thompson5910
    • one year ago
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    b-a = 7-5 = 2

  9. jim_thompson5910
    • one year ago
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    \[\Large \Delta x =\frac{b-a}{n}\] \[\Large \Delta x =\frac{7-5}{n}\] \[\Large \Delta x =\frac{2}{n}\] we just leave n as it is

  10. anonymous
    • one year ago
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    Ohhhhhh Makes much more sense

  11. anonymous
    • one year ago
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    So how would you finish it off?

  12. jim_thompson5910
    • one year ago
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    what would xi be ?

  13. jim_thompson5910
    • one year ago
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    \[\Large x_i = a+i*\Delta x\] \[\Large x_i = ???\]

  14. anonymous
    • one year ago
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    \[5+\frac{ 2 }{ n }i ?\]

  15. jim_thompson5910
    • one year ago
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    yep or \[\Large 5 + \frac{2i}{n}\] the two are equivalent

  16. jim_thompson5910
    • one year ago
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    A = exact area under the curve from x = 5 to x = 7 \[\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}f(x_i)*\Delta x\] \[\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}f\left(5+\frac{2i}{n}\right)*\frac{2}{n}\] \[\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}\left(5+\frac{2i}{n}\right)^4*\frac{2}{n}\]

  17. jim_thompson5910
    • one year ago
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    where \[\Large f(x) = x^4\]

  18. anonymous
    • one year ago
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    So that's the final answer?

  19. jim_thompson5910
    • one year ago
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    yeah

  20. anonymous
    • one year ago
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    Last question, why'd you put the limit there?

  21. jim_thompson5910
    • one year ago
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    well what the summation does is add up the rectangles (n of them) as you let n approach infinity, the rectangles get smaller and smaller like you see in this animation http://gimyuen.com/wp-content/uploads/2013/12/riemann_integral.gif

  22. jim_thompson5910
    • one year ago
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    so as n --> infinity, the approximate area gets closer and closer to the true exact area

  23. anonymous
    • one year ago
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    OK, I have two similar to this, can I ask you to check over my work?

  24. jim_thompson5910
    • one year ago
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    sure

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