1. Find the center mass of the region between y=x^2 and y=x^3 in the first quadr

Question

1. Find the center mass of the region between y=x^2 and y=x^3 in the first quadrant

and also the volume rotated about y-axis..

2. Find the exact length of the curve { x=1+sin(t); y=2-cos(t) } from t=0 to t=2pi.

3. Find the area between 1 and infinity for the curve, f(x)=1/x^2.

4. Given the initial value y(0)=1 for problem y'=y-1,

use Eulers method to sketch the solution using steps of 0.1.

Please go at least 5 steps foward!

5. For parametric curve x=t^2-12*sqrt(t), y=t^2-sin(t), find the first derivative and simplify.

6.Determine the surface area of the solid obtained by rotating the curve by y= square roots of (9-x^2) from x=-2 to x=2 about the x-axis.

7. A dam in the shape of a trapezoid with height 60 m and the lower base 80 m, upper base 20 m in length is filled to the top.Find the force on this dam.

Explanation / Answer

( 3 )

f(x) = 1/x^2

1/x^2 dx ( from 1 to infinity ) ==> -1/x ( from 1 to infinity ) ==. 1

( 5 )x=t^2-12*sqrt(t), y=t^2-sin(t)

dx/dt ==> 2t - 12/2sqrt(t) dy/dt == 2t - cost

==> 2t - 6/sqrt(t)

dy/dx = dy/dt .dt/dx ==>  2t - cost / 2t - 6/sqrt(t)

===> sqrt(t) ( 2t - cost ) ( t3/2 + 3 )/2 (t3 - 9)

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