What is the volume of the solid whose cross- sections are circular disks perpendicular to the x-axis and with diameters on the region bounded by the curves y = x and y=x^2?

the volume of the solid is

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 = ( x - x² )²

So, our volume will be;

V = ₀∫¹ A(x) dx

we substitute

= ₀∫¹ ( 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 ( x² + x⁴ - 2x³ ) dx  

= + -

= [ + - ]

= [ ]

V =

Therefore, the volume of the solid is