How do you use limits to find the area between the curve #y=x^2-x+1# and the x axis from [0,3]?
The area is
Therefore, the area is
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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:
- Define the function representing the curve: (y = x^2 - x + 1).
- Determine the limits of integration, which are (x = 0) and (x = 3).
- Set up the definite integral for the area: (\int_{0}^{3} (x^2 - x + 1) , dx).
- Integrate the function with respect to (x).
- Evaluate the definite integral using the limits of integration.
- 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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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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