Answer: look at explanation
Explanation:
To find the area between the two parabolas, we need to first find their intersection points. Let's start with the first parabola:
y² = 4(3+1)(x + 3 + 1) = 16(x + 4)
y = ± 4√(x + 4)
Now, let's work on the second parabola:
y² = 4(72 +1)32 +1 – x) 1+ 1/3
y² = 4(73)^(1/3) - 4x^(1/3) + 1
y = ± √(4(73)^(1/3) - 4x^(1/3) + 1)
We can set the two equations equal to each other to find the intersection points:
4√(x + 4) = ± √(4(73)^(1/3) - 4x^(1/3) + 1)
Squaring both sides and simplifying, we get:
16(x + 4) = 4(73)^(1/3) - 4x^(1/3) + 1
20x^(1/3) + 16x - 292 = 0
Let u = x^(1/3), then we have:
20u³ + 16u³ - 292 = 0
u³ = 73/5
u = (73/5)^(1/3)
x = u³ = 73/5
Now we can integrate to find the area between the parabolas:
A = ∫(from x=0 to x=73/5) [(4(73)^(1/3) - 4x^(1/3) + 1) - (4(x + 4))] dx
A = ∫(from x=0 to x=73/5) [4(73)^(1/3) - 4x^(1/3) - 4x - 15] dx
A = [4(73)^(1/3)x - 4(3/4)x^(4/3) - 2x² - 15x] (from x=0 to x=73/5)
A = 84(73)^(1/3)/5 - 292/5 - 438(73/5)^(4/3)/25 - 621/2
Therefore, the area between the parabolas is approximately 449.428 square units.