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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
2. 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 $\Large \Delta x =\frac{b-a}{n}$ $\Large x_i = a+i*\Delta x$

3. 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

4. jim_thompson5910

if a = 5 and b = 7, what is delta x equal to?

5. jim_thompson5910

any ideas?

6. anonymous

Wouldn't it equal 0? b is 7, a is 5, and n is infinity?

7. anonymous

@jim_thompson5910

8. jim_thompson5910

b-a = 7-5 = 2

9. 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

10. anonymous

Ohhhhhh Makes much more sense

11. anonymous

So how would you finish it off?

12. jim_thompson5910

what would xi be ?

13. jim_thompson5910

$\Large x_i = a+i*\Delta x$ $\Large x_i = ???$

14. anonymous

$5+\frac{ 2 }{ n }i ?$

15. jim_thompson5910

yep or $\Large 5 + \frac{2i}{n}$ the two are equivalent

16. jim_thompson5910

A = exact area under the curve from x = 5 to x = 7 $\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}f(x_i)*\Delta x$ $\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}f\left(5+\frac{2i}{n}\right)*\frac{2}{n}$ $\Large A = \lim_{n \to \infty} \sum_{i=1}^{n}\left(5+\frac{2i}{n}\right)^4*\frac{2}{n}$

17. jim_thompson5910

where $\Large f(x) = x^4$

18. anonymous

So that's the final answer?

19. jim_thompson5910

yeah

20. anonymous

Last question, why'd you put the limit there?

21. 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

22. jim_thompson5910

so as n --> infinity, the approximate area gets closer and closer to the true exact area

23. anonymous

OK, I have two similar to this, can I ask you to check over my work?

24. jim_thompson5910

sure

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