Answer :
The exact area of the region bunded by the given equations is 21.33 units.
A bounded region has either a bundary or some set of or constraints placed upon them. In other words a bounded shape cannot be an infinitely large area. A bounded anything has to be able to be contained along some parameters.
We have given that,
x² = y³ --------- (1)
and
x = 2y ---------- (2)
squaring this equation we get,
x² = 4y²
from equation (1),
y³ = 4y²
y³ - 4y² = 0
y²(y - 4) = 0
y² = 0 or y-4 = 0
y = 0 or y = 4
These will be our bounds of integration.
Therefore our expression of area will be,
[tex]A =[/tex] [tex]\int\limits^4_0 {y^{3} - 4y^{2} } \, dy[/tex]
A = [ [tex]\frac{y^{4} }{4}[/tex] - 4 [tex]\frac{y^{3} }{3}[/tex] ]₀⁴
= [ [tex]\frac{4^{4} }{4}[/tex] - 4 [tex]\frac{4^{3} }{3}[/tex] - 0]
= 64 - 85.33
A = -21.33
i.e A = 21.33
Therefore area of bounded region is 21.33 units.
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