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stiks02 [169]
3 years ago
10

What is the volume of the solid whose cross- sections are circular disks perpendicular to the x-axis and with diameters on the r

egion bounded by the curves y = x and y=x^2?
Mathematics
1 answer:
12345 [234]3 years ago
6 0

Answer:

the volume of the solid is \frac{\pi }{120}

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 = \frac{\pi }{4}( x - x² )²

So, our volume will be;

V = ₀∫¹ A(x) dx

we substitute

= ₀∫¹ \frac{\pi }{4}( 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 \frac{\pi }{4}( x² + x⁴ - 2x³ ) dx  

= \frac{\pi }{4} [ \frac{x^3}{3} + \frac{x^5}{5} - \frac{2}{4} x^4 ]^1_0

= \frac{\pi }{4}[ \frac{1}{3} + \frac{1}{5} - \frac{2}{4} ]

= \frac{\pi }{4}[ \frac{1}{30} ]

V = \frac{\pi }{120}

Therefore, the volume of the solid is \frac{\pi }{120}

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