Question

please do it my homework as clear as u can !

Answer 1

y = √x  and  y = x³

these curve meet  when  x³ = √x
       =>  x = 0  or  1

y = √x  is higher than  x³  between x= 0 and x = 1.

areas under the curves / enclosed by the curves :

$A= \int\limits^1_{x=0} {(y_2-y_1)} \, dx\\\\=\int\limits^1_0 {(\sqrt{x}-x^3)} \, dx\\\\=[\frac{2}{3}x^{\frac{3}{2}}-\frac{1}{4}x^4}]_0^1\\\\=\frac{2}{3}-\frac{1}{4}\\\\=\frac{5}{12}$

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at any x, the radius of the circular cross section of the paraboloid is y..

Volume of an infinitesimal  volume of a disc of radius y  and  thick ness   dx
       = π y²  dx

   the equation of the curve is     y = √x      or  x =  y²

Volume of the paraboloid :

$V= \int\limits^b_0 {\pi\ y^2} \, dx\\\\= \int\limits^b_0 {\pi\ x} \, dx \\\\=\frac{\pi}{2}\ [x^2]_0^b\\\\=\pi b^2/2$

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R = 50.0 ft 
A = 2 π / 3 radians = 120 deg. = 2π/3 radians

Height raised by, when the wheel turns by 90 deg.  =  1 radius. = 50 ft.
          as the person is at the level of the center of the wheel.
height raised by, when the wheel turns by 30 deg more =  R Sin (2π/3 - π/2) = R/2 = 25 ft.
The height above the ground = 25 + 50 = 75 feet..
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Angular speed = ω = angle turned / time duration = [ 2 π /3 ] / 30 sec
             = π / 45  radians / sec
             = 0.0698 radians / sec