Answer:
the volume of the solid is [tex]\frac{\pi }{120}[/tex]
Step-by-step explanation:
Given the data in the question;
y = x and y = x²
so
x = x²
x - x² = 0
Radius will be; r = ( x - x² )/2
Area of the circular disk πr² = π[ ( x - x² )/2 ]²
A = [tex]\frac{\pi }{4}[/tex]( x - x² )²
So, our volume will be;
V = ₀∫¹ A(x) dx
we substitute
= ₀∫¹ [tex]\frac{\pi }{4}[/tex]( x - x² )² dx
{ lets expand; ( x - x² )² = x(x-x²) -x²(x-x²) = x² - x³ - x³ + x⁴ = x² + x⁴ - 2x³ }
so we have;
= ₀∫¹ [tex]\frac{\pi }{4}[/tex]( x² + x⁴ - 2x³ ) dx
= [tex]\frac{\pi }{4}[/tex] [tex][[/tex] [tex]\frac{x^3}{3}[/tex] + [tex]\frac{x^5}{5}[/tex] - [tex]\frac{2}{4} x^4[/tex] [tex]]^1_0[/tex]
= [tex]\frac{\pi }{4}[/tex][ [tex]\frac{1}{3}[/tex] + [tex]\frac{1}{5}[/tex] - [tex]\frac{2}{4}[/tex] ]
= [tex]\frac{\pi }{4}[/tex][ [tex]\frac{1}{30}[/tex] ]
V = [tex]\frac{\pi }{120}[/tex]
Therefore, the volume of the solid is [tex]\frac{\pi }{120}[/tex]
