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

Answer 1

#64/3#

First find the integral of the function: #int(8+2y-y^2)dy = -1/3y^3+y^2 +8y#
There are three intervals that you should solve that is #[-3,-2],[-2,0],[0,1]#. The problem of plugging #-3# directly into the integral would subtract the area the left of the #-2# root.
(1)#int[-2,0]# #=8(-2)+(-2)^2-1/3(-2)^3=abs(-28/3)=28/3#
(2)#int[-3,-2]=int[-3,0]-int[-2,0]# #=8(-3)+(-3)^2-1/3(-3)^3-(-28/3)=-6-(-28/3)=10/3#
(3)#int[0,1]# #=8(1)-(1)^2-1/3(1)^3=26/3#
Adding all together: #28/3+10/3+26/3=64/3#
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Answer 2

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

  1. 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).
  2. Next, find the points of intersection between the curve and the (y)-axis by setting (x = 0) and solving for (y).
  3. After identifying these points, sketch the curve and the lines to understand the region bounded by them.
  4. 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.
  5. Subtract any additional areas enclosed by the curve and the lines (y = -1) and (y = 3) outside the main region.
  6. The result of the integral gives the area bounded by the curve, the (y)-axis, (y = -1), and (y = 3).
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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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