Find the number #b# such that the line y=b divides the region...?

Find the number #b# such that the line y=b divides the region bounded by the curves #y=x^2# and #y=4# into two regions with equal area.

Answer 1

Find the area of the region first.

The area of the region is given by #int_-2^2(4-x^2)dx = 2int_0^2(4-x^2)dx# #= 2(4x-x^3/3)_0^2# #= 32/3#
Second, #y = b# intersects the curve #y = x^2# when #x = +-sqrtb#. Third, we want to find b such that #int_-sqrtb^sqrtb(b-x^2)dx = 16/3# This will occur if and only if #int_0^sqrtb(b-x^2)dx = 8/3# Integrate: #int_0^sqrtb(b-x^2)dx =(bx-x^3/3)_0^sqrtb# #= bsqrtb-(bsqrtb)/3# #= (2bsqrtb)/3#
Now set this equal to #8/3#. #(2bsqrtb)/3 = 8/3# #bsqrtb = 4# #b^(3/2) = 4# #b = root(3)(16)# or #b = 2root(3)(2)#
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Answer 2

To find the number (b) such that the line (y = b) divides the region enclosed by the curves (y = x^2 - 4x + 3) and (y = 3 - x^2) into two regions of equal area, we need to set up and solve an equation.

Let's denote the points of intersection of the two curves as (x_1) and (x_2). Then, (b) will be the midpoint of the y-values of these intersection points.

  1. Find the intersection points: Set the two equations equal to each other and solve for (x): [x^2 - 4x + 3 = 3 - x^2] Solve for (x) to find the intersection points.

  2. Calculate the midpoint of the y-values of the intersection points: Once you have (x_1) and (x_2), find the corresponding y-values for each curve. Then, find the midpoint of these y-values, which will be (b).

This value of (b) will be the number such that the line (y = b) divides the region enclosed by the curves into two regions of equal area.

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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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