what is the area of the region bounded by the curves y=x^2 , y=8-x^2 , 4x-y+12=0

After making a diagram, I cannot tell from the question if you mean the smaller part in quadrant II or the larger part of the region between the two parabolas.

the two parabolas intersect at (2,4) and (-2,4)

and the line intersects the y=x^2 at (-2,4) and (6,36)
and is tangent to the downwards parabola at (-2,4)

I don't see an area bounded by all three.
So the area is zero.

Check your question.

To find the area of the region bounded by the given curves, we need to find the points where the curves intersect and then calculate the area between those points.

1. Begin by finding the points of intersection between the curves:

- The first two curves are y = x^2 and y = 8 - x^2. Set them equal to each other and solve for x:

x^2 = 8 - x^2
2x^2 = 8
x^2 = 4
x = ±2

- The third curve is the equation of a line, 4x - y + 12 = 0. Rewrite it in terms of y:

y = 4x + 12

Now we will find the points where this line intersects with the curves:

Substituting y = 4x + 12 into the equation of the first curve:
x^2 = 4x + 12
x^2 - 4x - 12 = 0

Solving this quadratic equation, we get two solutions: x = -2 and x = 6.

Substituting y = 4x + 12 into the equation of the second curve:
8 - x^2 = 4x + 12
x^2 + 4x - 4 = 0

Again, solving this quadratic equation, we find two solutions: x = -2 and x = -6.

Therefore, the points of intersection are (-2, 4), (-2, 4), (2, 4), and (-6, 8).

2. Next, we need to determine the limits of integration for calculating the area. The region is bounded between the curves y = x^2 and y = 8 - x^2.

To determine the limits, we need to find the x-values where the curves intersect. In this case, the intersection points are x = -2 and x = 2.

So, the limits of integration will be x = -2 to x = 2.

3. Finally, we can calculate the area between the curves using definite integration:

The area (A) between the curves can be calculated using the following integral:

A = ∫[lower limit, upper limit] (top curve - bottom curve) dx

In this case, the top curve is y = 8 - x^2, and the bottom curve is y = x^2. Therefore, the integral becomes:

A = ∫[-2, 2] [(8 - x^2) - x^2] dx

Evaluating this integral will give you the area of the region bounded by the given curves.

Note: Since this process involves solving quadratic equations and performing integrals, the exact value of the area may be complex. You may need to use appropriate techniques or numerical methods to evaluate the integral or approximate the area.