anonymous
  • anonymous
Write the Riemann sum to find the area under the graph of the function f(x) = x4 from x = 5 to x = 7.
Calculus1
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
https://i.gyazo.com/5703a9ba56f6d799fdd797a328f5c247.png
jim_thompson5910
  • jim_thompson5910
|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 \[\Large \Delta x =\frac{b-a}{n}\] \[\Large x_i = a+i*\Delta x\]
anonymous
  • anonymous
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

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jim_thompson5910
  • jim_thompson5910
if a = 5 and b = 7, what is delta x equal to?
jim_thompson5910
  • jim_thompson5910
any ideas?
anonymous
  • anonymous
Wouldn't it equal 0? b is 7, a is 5, and n is infinity?
anonymous
  • anonymous
@jim_thompson5910
jim_thompson5910
  • jim_thompson5910
b-a = 7-5 = 2
jim_thompson5910
  • jim_thompson5910
\[\Large \Delta x =\frac{b-a}{n}\] \[\Large \Delta x =\frac{7-5}{n}\] \[\Large \Delta x =\frac{2}{n}\] we just leave n as it is
anonymous
  • anonymous
Ohhhhhh Makes much more sense
anonymous
  • anonymous
So how would you finish it off?
jim_thompson5910
  • jim_thompson5910
what would xi be ?
jim_thompson5910
  • jim_thompson5910
\[\Large x_i = a+i*\Delta x\] \[\Large x_i = ???\]
anonymous
  • anonymous
\[5+\frac{ 2 }{ n }i ?\]
jim_thompson5910
  • jim_thompson5910
yep or \[\Large 5 + \frac{2i}{n}\] the two are equivalent
jim_thompson5910
  • jim_thompson5910
A = exact area under the curve from x = 5 to x = 7 \[\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}f(x_i)*\Delta x\] \[\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}f\left(5+\frac{2i}{n}\right)*\frac{2}{n}\] \[\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}\left(5+\frac{2i}{n}\right)^4*\frac{2}{n}\]
jim_thompson5910
  • jim_thompson5910
where \[\Large f(x) = x^4\]
anonymous
  • anonymous
So that's the final answer?
jim_thompson5910
  • jim_thompson5910
yeah
anonymous
  • anonymous
Last question, why'd you put the limit there?
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
so as n --> infinity, the approximate area gets closer and closer to the true exact area
anonymous
  • anonymous
OK, I have two similar to this, can I ask you to check over my work?
jim_thompson5910
  • jim_thompson5910
sure

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