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Find the area of the region between the curves y=4x^2 −4 and y=−5x^2 +5 from x=−1 to x=1 The area is _______

(Simplify your answer. Type an integer or a fraction.)

User Sunil Dora
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To find the area between two curves, we will use the concept of definite integrals.

Given two functions, y = 4x^2 - 4 and y = -5x^2 + 5, we can find the area between these two curves from x = -1 to x = 1 by finding the definite integral of the absolute value of their difference over the interval from -1 to 1.

First, subtract y = -5x^2 + 5 from y = 4x^2 - 4, yielding 9x^2 - 9.

Now, since we are finding the area, we need the absolute value of this function. This is necessary because the area cannot be negative.

The function is symmetric around the y-axis, so we can integrate from 0 to 1 and multiply by 2 to get the result from -1 to 1.

Now the problem is simplified to finding

2 * ∫ from 0 to 1 |9x^2 - 9| dx.

The absolute value function will change the sign of the values that make 9x^2 - 9 negative.

Since we are working in the interval from 0 to 1, 9x^2 - 9 is always negative. Thus, we remove the absolute value bars and change the sign of the function inside to get

2 * ∫ from 0 to 1 (9 - 9x^2) dx.

Now we need to calculate a definite integral.

∫ from 0 to 1 (9 - 9x^2) dx = [9x - 3x^3] from 0 to 1 = 9*1 - 3*1^3 - (9*0 - 3*0^3) = 6.

Remember, this is only half of the solution because we halved the interval in order to get rid of the absolute value.

So, multiplying 6 by 2, we get a final solution of 12.

Therefore, the area of the region between the curves y = 4x^2 - 4 and y = -5x^2 + 5 from x = -1 to x = 1 is 12 square units.

User Kundan Bora
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