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tavale21

  • 2 years ago

approximate the area under each curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the right-hand endpoints of the subintervals. f(x)=x^3 from x=0 to x=3; 3 subintervals

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  1. wio
    • 2 years ago
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    |dw:1353031457071:dw|

  2. wio
    • 2 years ago
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    You basically want to find the area of the three rectangles. Make sense?

  3. tavale21
    • 2 years ago
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    sort of? how would i find the area's ?

  4. wio
    • 2 years ago
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    Well, for one, what is the area of a rectangle?

  5. tavale21
    • 2 years ago
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    is it from 0 to 3?

  6. wio
    • 2 years ago
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    I'm talking about in the general sense

  7. tavale21
    • 2 years ago
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    that's all that the book has. I looked at another problem and it is f(x) = 4x-x^2 from x=0 to x=2; 2 subintervals and the answer was 7 square units. how did they get that?

  8. wio
    • 2 years ago
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    Okay, I just wanted to say, the area of a rectangle is length times width.

  9. wio
    • 2 years ago
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    So our first task is to find the width of the rectangles.

  10. wio
    • 2 years ago
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    We will split up the interval equally into sub intervals... So we know with n subintervals on the interval [a,b], the width w is:\[ nw = b-a \implies w=\frac{b-a}{n} \]In this case, b = 3, a =0, and n = 3

  11. wio
    • 2 years ago
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    So what is the width of each rectangle?

  12. tavale21
    • 2 years ago
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    ok

  13. tavale21
    • 2 years ago
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    3

  14. wio
    • 2 years ago
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    That is the width of the entire interval, not the width of each subinterval/rectangle

  15. wio
    • 2 years ago
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    \[ \frac{3-0}{3} = 1 \]

  16. tavale21
    • 2 years ago
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    oh ok. I see now b-a/n=1

  17. wio
    • 2 years ago
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    Now, to find the length of the interval is a bit tricky.

  18. wio
    • 2 years ago
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    We need to pick some x in each subinterval. The length is then f(x).

  19. wio
    • 2 years ago
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    But which x do we pick? It depends on the problem. This problem says that you want to pick the right-hand endpoints, so we want to pick the x such that x is the highest value of x for that subinterval.

  20. wio
    • 2 years ago
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    The first interval is from a to a+w, in this case 0 to 1. Since 1 is highest, right hand side, we use f(1) for the length of the interval.

  21. wio
    • 2 years ago
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    So the first interval is \(1*f(1) = 1 * 1^3 = 1\)

  22. wio
    • 2 years ago
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    The second interval is between a+w and (a+w)+w

  23. wio
    • 2 years ago
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    So the highest point is a+w+w = 0+1+1 = 2

  24. wio
    • 2 years ago
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    So the second subinterval area is \(1*f(2) = 1*2^3=1*8=8\)

  25. wio
    • 2 years ago
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    Do you think you can do the third sub interval?

  26. tavale21
    • 2 years ago
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    can you tell me the steps sorry im not good at this?

  27. wio
    • 2 years ago
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    |dw:1353032610122:dw|

  28. wio
    • 2 years ago
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    Okay, first step is to find with width of the rectangles. Next step is to find the length of each rectangle.

  29. wio
    • 2 years ago
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    Then you add the areas of all the rectangles. Do you at least understand the high level idea of what we are doing?

  30. tavale21
    • 2 years ago
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    are we adding the last f(3) to 8

  31. tavale21
    • 2 years ago
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    the answer f(3) * 8

  32. wio
    • 2 years ago
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    Nope. All I did was show that rectangle 1 has area 1, and rectangle 2 has area 8

  33. wio
    • 2 years ago
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    So you need to find width times f(3) to find the area of rectangle 3. Then you add up the areas of all rectangles.

  34. tavale21
    • 2 years ago
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    1*3^3=27

  35. tavale21
    • 2 years ago
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    27+8+1= 36

  36. tavale21
    • 2 years ago
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    can you explain how the left hand works? to the same problem? it makes sense now? thank you!!!

  37. wio
    • 2 years ago
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    |dw:1353033340267:dw|

  38. wio
    • 2 years ago
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    For the right hand, we use f(a+w), f(a+2w), f(a+3w), ... , f(a+(n-1)w), f(a+nw) For the left hand, we use f(a), f(a+w), f(a+1w), ... , f(a+(n-2)w), f(a+(n-1)w)

  39. tavale21
    • 2 years ago
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    ok

  40. wio
    • 2 years ago
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    How do you understand how I'm getting them?

  41. tavale21
    • 2 years ago
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    from the picture you drawn up you use the rectangles starting from the left to the right.

  42. wio
    • 2 years ago
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    You will learn later, a method to do this with infinite subintervals. It's called a definite integral.

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