## jennychan12 Group Title Integral Question. See below, one year ago one year ago

1. Dido525 Group Title

O_o .

2. jennychan12 Group Title

If $\int\limits_{0}^{6} (x^2-2x+2)dx$ is approximated by 3 inscribed rectangles of equal width, then the approximation is... 1) 24 B) 26 C) 28 D)76 E) 48

3. jennychan12 Group Title

sorry typed something wrong. and that 1 is supposed to be an A

4. jennychan12 Group Title

so i got the integrated area to be 48, so answers either a b or c

5. Dido525 Group Title

Wait.

6. Dido525 Group Title

Wait, what kind of rectangles? Left, right, or midpoint?

7. jennychan12 Group Title

the problem doesnt say. i'm assuming left

8. jennychan12 Group Title

cuz it says inscribed

9. Dido525 Group Title

Well you are right. It's either A, B or C :D . Now can you do it out? :) .

10. Dido525 Group Title

$\Delta(x)=\frac{ 6-0 }{ 3 }$

11. jennychan12 Group Title

i'm confused my friend says to use riemann sum. what's the formula for that?

12. Dido525 Group Title

$\Delta(x) = 2$

13. Dido525 Group Title

In this particular example, a Riemann sum is basically the sum of the individual rectangles.

14. Dido525 Group Title

When you learn about definite intergals you will have a good idea about what they actually mean.

15. jennychan12 Group Title

is it $\frac{ b-a }{ n } \sum_{i = 1}^{n} f(x)$ ??

16. Dido525 Group Title

No wait, that's right.

17. Dido525 Group Title

So yeah you get the idea. It's the sum of all the rectangles multiples by the change in x,(i.e the width) .

18. jennychan12 Group Title

that's what i did but i got 28 and the answer's 26

19. Dido525 Group Title

Then you have to use diffrent kinds of rectangles because your answer is correct.

20. jennychan12 Group Title

26 or 28 ?

21. Dido525 Group Title

It's 26.

22. Dido525 Group Title

But if you use left rectangles then it's 28.

23. jennychan12 Group Title

why? sorry can u explain?

24. jennychan12 Group Title

oh.

25. jennychan12 Group Title

so right hand it's 26 or..?

26. Dido525 Group Title

So this is a very stupid question because it dosen't specify the type of rectangles.

27. Dido525 Group Title

Well right hand is 76 actually.

28. Dido525 Group Title

But it dosen't specify what type of rectangle. Left, right or midpoint.

29. jennychan12 Group Title

where'd the 26 come from? midpoint?

30. Dido525 Group Title

No, midpoint is 46.

31. Dido525 Group Title

But shall I explain left and right rectangles?

32. Dido525 Group Title

If you use left, you get 28. If you use right you get 76.

33. Dido525 Group Title

Shall I explain?

34. jennychan12 Group Title

wait so where did the 26 come from. please explain. i'm so lost. :(

35. Dido525 Group Title

36. jennychan12 Group Title

cuz this other person used 2(f(1)+f(2)+f(4)) and got 26 but i don't understand where the f(1) came from...

37. jennychan12 Group Title

oh.

38. Dido525 Group Title

Wait hold on.

39. Dido525 Group Title

Yeah, 28 should be correct. I disagree with the answer.

40. Dido525 Group Title

If you use 3 subintervals you CANNOT get 26 in anyway.

41. Dido525 Group Title

The formula would be (2)(f(0)+f(2)+f(4))

42. jennychan12 Group Title

yeah i did that... and got 28

43. Dido525 Group Title

Which is absolutely correct.

44. Dido525 Group Title

It CANNOT be 26 if you use left rectangles.

45. Dido525 Group Title

if you use right rectangles it's (2)(f(2)+f(4)+f(6)) which is 76.

46. jennychan12 Group Title

yeah i got 28 and 76 but i chose 28. so my teacher is wrong. hmph.

47. Dido525 Group Title

Yep, you teacher is wrong.

48. jennychan12 Group Title

thanks for clarifying. :)

49. Dido525 Group Title

In reality the exact value of that integral is 48 but you will learn that later.

50. jennychan12 Group Title

yeah i know. i calculated it.

51. Dido525 Group Title

Kk.

52. yun2thejae Group Title

Good 'ol Riemann's screwing with everybody's head.

53. Dido525 Group Title

I just hope I don't have to do it after my first year in uni :/ . AP calc helped so much though XD .

54. jennychan12 Group Title

ohhhhhhh i think i got it they used average value

55. yun2thejae Group Title

LOL

56. oldrin.bataku Group Title

$$A\simeq2(2^2-2(2)+2+4^2-2(4)+2+6^2-2(6)+2)\\\ \ \ =2(4-4+2+16-8+2+36-12+2)\\\ \ \ =2(38)=76$$ (using right end-points)$$A\simeq2(0^2-2(0)+2+2^2-2(2)+2+4^2-2(4)+2)=2(2+4-4+2+16-8+2)\\\ \ \ =2(14)=28$$ (using left end-points)$$A\simeq2(1^2-2(1)+2+3^2-2(3)+2+5^2-2(5)+2)\\\ \ \ =2(1-2+2+9-6+2+25-20+2)\\\ \ \ =2(13)=26$$ (using midpoints)

57. Dido525 Group Title

^ Reason why they they should specify the question.

58. jennychan12 Group Title

my friend just assumed that inscribed meant left end points

59. Dido525 Group Title

@oldrin.bataku : Please don't just give the answer next time. The point of this place is to help other people learn. Not give them the answer.

60. Dido525 Group Title

@jennychan12 : That's a flase assumption.

61. Dido525 Group Title

false*

62. Dido525 Group Title

@oldrin.bataku : If you use midpoints, it's 46 not 26.

63. yun2thejae Group Title

Lol Oldrin and i have been spending plenty of time explaining problems to different users. I would give him a break xD. I was bored and decided to use this site for the first time today and, oddly enough, I can't stop attempting to solve problems. Probably because I don't want to forget my math whilst im on winter break.

64. Dido525 Group Title

I know but it's just that's it's against the TOS so I try not to. I make that mistake too. It's cool ^_^ .

65. oldrin.bataku Group Title

@Dido525 you're right, 2(5) = 10 so it's actually 2(23) = 46 for the approximate area.

66. oldrin.bataku Group Title

If it requires inscribed rectangles then the approximation must use left end-point... mid- and right- will not be entirely under the parabola.

67. Dido525 Group Title

That's why we have the trapozoidal rule ;) . Ans simpsons rule :P .

68. Dido525 Group Title

and*

69. Dido525 Group Title

But yes @oldrin.bataku is correct. If they MUST be inscribed then you MUSt use left rectangles.

70. Dido525 Group Title

MUST*

71. oldrin.bataku Group Title

@Dido525 if you go into a Scientific Computation course you'll learn about plenty more numerical methods e.g. Runge-Kutta or Adams-Moulton :-p Euler's and Verlet are really common for physics problems.

72. yun2thejae Group Title

Conversation is too much for this PoSci major xD.

73. Dido525 Group Title

I looked at Runge-kutta but it makes no sense XD . I am going much deeper into math next year so hopefully then :) .

74. Dido525 Group Title

75. jennychan12 Group Title

it's aiight.

76. agent0smith Group Title

Nobody drew a damn diagram :P

77. Dido525 Group Title

Sigh...

78. yun2thejae Group Title

Diagram was drawn in the other thread.

79. jennychan12 Group Title

^ haha yes

80. Dido525 Group Title

|dw:1355899568667:dw|

81. Dido525 Group Title

Something like that.

82. jennychan12 Group Title
83. agent0smith Group Title

that's the same thread as this one...

84. yun2thejae Group Title

LOL

85. Dido525 Group Title

|dw:1355899651047:dw| That would be incorrect since it does not enclose the curve.

86. jennychan12 Group Title

yeah i got really confused and tried to do another one to resolve my confusion... :D cuz that other person made me confused

87. oldrin.bataku Group Title

|dw:1355899646424:dw|

88. Dido525 Group Title

|dw:1355899717852:dw|

89. Dido525 Group Title

If you use midpoints it still dosen't enclose the curve fully.

90. agent0smith Group Title

Inscribed rectangles *should* mean rectangles within the curve, but it sounds like that didn't work, so the damn question is broken.

91. Dido525 Group Title

Damn question.

92. Dido525 Group Title

It's is 28 though without a doubt.

93. oldrin.bataku Group Title

94. jennychan12 Group Title

yeah

95. agent0smith Group Title

inscribed rectangles would give 26, but i feel like that was already covered in this mess somewhere

96. yun2thejae Group Title

You haven't done "work" problems in your calc class yet have you?

97. agent0smith Group Title

Wait, why isnt the answer 26? 3 inscribed rectangles = area of 26.

98. jennychan12 Group Title

what do u mean? kinetic energy? i do that in physics....

99. jennychan12 Group Title

@agent0smith can u explain???

100. yun2thejae Group Title

there are "work" problems you use calculus for. Pray that you do not have to do them xD

101. Dido525 Group Title

If you use 3 inscribed rectangles it's 28 not 26.

102. yun2thejae Group Title

The previous thread came up with 26. this one says 28. let's meet half way and say it's 27...oh wait.

103. jennychan12 Group Title

lol

104. agent0smith Group Title

The height of the first is 1, length is 2 height of the second is 2, length is 2 height of the third is 10, length is 2. 2+4+20

105. agent0smith Group Title

How'd you get 28, @Dido525

106. jennychan12 Group Title

left endpoint is 28

107. Dido525 Group Title

=(2) * ( f(0) + f(2) + f(4) ) = (2) * ( 2 + 2 + 10 ) = 28

108. agent0smith Group Title

Inscribed have to be all within the curve, no edges jutting out: it's not f(0), it's f(1), the height of the first rectangle.

109. agent0smith Group Title

That is not an inscribed rectangle.

110. jennychan12 Group Title

that's what that other person did in that other thread 2(f(1)+f(2)+f(4)) = 26 but why would be f(1)??? where does the 1 even come from?

111. Dido525 Group Title

But f(0) is inscribed.

112. agent0smith Group Title

I'll draw it jenny. One sec.

113. Dido525 Group Title

I want to see this too.

114. agent0smith Group Title

dido, look at the graph: http://www.google.com/search?q=x%5E2-2x%2B2&aq=f&oq=x%5E2-2x%2B2&sugexp=chrome,mod=18&sourceid=chrome&ie=UTF-8 and then look where a rectangle at f(0) would fall... it's not inscribed. it goes over the curve.

115. Dido525 Group Title

Isn't that the same thing as inscribed?

116. Dido525 Group Title

Inscribed meaning within.

117. agent0smith Group Title

No, inscribed means it falls completely within the curve, no jutting out over it.

118. jennychan12 Group Title

okay. i understand that f(1) = 1 but.... oh wait. i see where you're going with this....

119. Dido525 Group Title

|dw:1355900439887:dw|

120. Dido525 Group Title

But rectangles will never cover a curve fully, it's impossible.

121. Dido525 Group Title

Like, it covers it fully but it's an overestimate.

122. agent0smith Group Title

123. Dido525 Group Title

Ohh shoot! O_o .

124. agent0smith Group Title

Those are inscribed rectangles, they underestimate the area greatly of course.

125. Dido525 Group Title

Ahh, NOW this makes sense.

126. agent0smith Group Title

hahah :)

127. Dido525 Group Title

But if it's an underestimate, it's not inscribed is it?

128. jennychan12 Group Title

ok yeah i see it too.

129. agent0smith Group Title

I googled inscribed rectangles just to be sure.

130. Dido525 Group Title

Wait, why did you do a midpoint for the first rectangle?

131. Dido525 Group Title

132. agent0smith Group Title

Because that's the only way to have it inscribed. http://en.wikipedia.org/wiki/Inscribed_figure

133. Dido525 Group Title

NOW this all makes sense :D .

134. agent0smith Group Title

Inscribed rectangles can be either left or right endpoint

135. agent0smith Group Title

or midpoint

136. agent0smith Group Title

@jennychan12 you see how it's 26 now? the height of the first is f(1), next is f(2), then f(4)

137. Dido525 Group Title

I would give you my medal if I could :P .

138. agent0smith Group Title

haha :) this whole confusion was all over the meaning of the word inscribed :D

139. Dido525 Group Title

Agreed! :D . But hey, this might be n my final so it was good for me too :) .

140. agent0smith Group Title

haha good :) hopefully this clears things up Jenny, before you went and told your teacher he/she was wrong because a bunch of people on the internet said so! :D

141. Dido525 Group Title

She is offline :/ .

142. Dido525 Group Title

@jada1701877 : You should have given that to agent.

143. jennychan12 Group Title

thanks so much! and yeah, i went off at like 11 cuz i went to sleep. :)