# How do you find the area bounded by #x=8+2y-y^2#, the y axis, y=-1, and y=3?

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To find the area bounded by the curve (x = 8 + 2y - y^2), the (y)-axis, (y = -1), and (y = 3), follow these steps:

- First, determine the points of intersection between the curve and the vertical lines (y = -1) and (y = 3) by substituting these (y)-values into the equation of the curve and solving for (x).
- Next, find the points of intersection between the curve and the (y)-axis by setting (x = 0) and solving for (y).
- After identifying these points, sketch the curve and the lines to understand the region bounded by them.
- Calculate the definite integral of the absolute value of the curve's equation with respect to (y) from the lower bound (y = -1) to the upper bound (y = 3). This integral represents the area enclosed by the curve and the (y)-axis between the specified (y)-values.
- Subtract any additional areas enclosed by the curve and the lines (y = -1) and (y = 3) outside the main region.
- The result of the integral gives the area bounded by the curve, the (y)-axis, (y = -1), and (y = 3).

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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.

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