How do you use limits to find the area between the curve #y=x^2-x+1# and the x axis from [0,3]?

Answer 1

The area is #15/2=7.5#.

In general, for a "nice" (continuous) function #f# whose graph is above the #x#-axis for #a <=x<=b#, the area under the graph can be computed using limits as #\int_{a}^{b}f(x)dx=lim_{n->infty}\sum_{k=1}^{n}f(x_{k})\Delta x#, where #Delta x=(b-a)/n# and #x_{k}=a+k*Delta x#.
For the given situation, #a=0# and #b=3# so that #Delta x=3/n# and #x_{k}=0+k*3/n=(3k)/n#. We then get #f(x_{k})=((3k)/n)^2-(3k)/n+1=(9k^2)/n^2-(3k)/n+1#.
This leads to #sum_{k=1}^{n}f(x_{k})\Delta x=sum_{k=1}^{n}((27k^2)/n^3-(9k)/n^2+3/n)#
#=27/n^3*sum_{k=1}^{n}k^2-9/n^2 * sum_{k=1}^{n}k+3/n * sum_{k=1}^{n}1#.
Now #sum_{k=1}^{n}1=n#, #sum_{k=1}^{n}k=(n(n+1))/2#, and #sum_{k=1}^{n}k^2=(n(2n+1)(n+1))/6# (you can look these facts up...for example, see https://tutor.hix.ai ).
Thus, #sum_{k=1}^{n}f(x_{k})\Delta x#
#=(27n(2n+1)(n+1))/(6n^3)-(9n(n+1))/(2n^2)+3#

Therefore, the area is

#\int_{a}^{b}f(x)dx=lim_{n->infty}\sum_{k=1}^{n}f(x_{k})\Delta x#
#=54/6-9/2+3=15/2#
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Answer 2

To find the area between the curve (y = x^2 - x + 1) and the x-axis from (x = 0) to (x = 3) using limits, follow these steps:

  1. Define the function representing the curve: (y = x^2 - x + 1).
  2. Determine the limits of integration, which are (x = 0) and (x = 3).
  3. Set up the definite integral for the area: (\int_{0}^{3} (x^2 - x + 1) , dx).
  4. Integrate the function with respect to (x).
  5. Evaluate the definite integral using the limits of integration.
  6. Calculate the area.

(A = \int_{0}^{3} (x^2 - x + 1) , dx)

(A = \left[\frac{x^3}{3} - \frac{x^2}{2} + x\right]_{0}^{3})

(A = \left[\frac{3^3}{3} - \frac{3^2}{2} + 3\right] - \left[\frac{0^3}{3} - \frac{0^2}{2} + 0\right])

(A = \left[\frac{27}{3} - \frac{9}{2} + 3\right] - \left[0 - 0 + 0\right])

(A = (9 - 4.5 + 3) - (0))

(A = 7.5)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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