The area bounded by the curves is 104.53 unit².
How to calculate area between curves.
Given
5x² + y = 33 and x⁴ - y = 3
Rearrange the second equation to y = x⁴ - 3 then substitute this expression into the first equation:
5x² + (x⁴ - 3) = 33
Combine like terms:
x⁴ + 5x² - 36 = 0
Now, factor the quadratic:
(x² - 4)(x² + 9) = 0
This gives two sets of solutions:
x² - 4 = 0, x = +-2
x² + 9 = 0 has no real solutions.
So, the points of intersection occur at x = -2 and x = 2.
The area A between the curves can be found using the integral:
A = ∫₋₂²[(33-5x²) - (x⁴- 3)]dx
A = ∫₋₂²[(33-5x²- x⁴ + 3)]dx
Simplify further:
A = ∫₋₂²[(36-5x² - x⁴)]dx
Now, integrate with respect to x
A = [(36x -5/3x³ - 1/5x⁵)]₋₂²
Evaluate the expression at the upper and lower limits:
A = [(36(2) -5/3(2)³ - 1/5(2)⁵) - [(36(-2) -5/3(-2)³ - 1/5(-2)⁵)]
A = [(72 -40/3 - 32/5) - [( -72 + 40/3 + 32/5)
= 144 - 80/3 - 64/5
= 104.53 unit²