Geometry and Calculus Questions..?

Question

1. the base of a solid is the region in the xy plane bounded by the
parabolas y=x^2 ans x=^y2. find the volume of the solid if every cross
section perpendicular to the x-axis is a square with its base in the
xy plane.

2. given y=x^5 - 4x^3 + 5x
a. locate the exact or approximate intercepts
b. identify any axis of symmetry
c. determine the curves behavior at positive and negative infinity
d. use the first and second derivative to locate turning points

3. find the area common to r=1+cos@ and r=3^(1/2) sin@

4. specify the points of intersection of rcos@=1 and r=4cos@

5. tnagents are drawn to the ellipse x^2/a^2 + y^2/b^2 =1 and the
circle x^2 + y^2 = a^2 at points having the same abscissa. prove that
these tangents cross OX (a-axis) at the same point.

6. find the volume generated by rotating the region bounded by
(x-1)^2 + (y-2)^2 = 4 around
a. x axis
b. y -axis
c. x = 3
d. y = 4

[Difficulty]
I need help in one and five a lot!!

Answer 1

1) The side of the square is sqrt(x)-x^2 so the volume is
Int (0,1) pi* [sqrt(x)-x^2]^2 dx =pi* Int( x+x^4-2x^2 sqrt(x))dx=
(pi)x^2/2+x^5/5 -4/7 x^7/2 (0,1) = (1/2 +1/5 -4/7)*pi = 9/70*pi
2) x^5-5x^3 +5x=0 so x=0 and
x^4-5x^2+5=0 put x^2=z so z^2-5z+5=0 and z=((5+-sqrt(45))/2 As z can be only >=0
x= +- sqrt[ (5+sqrt(45))/2]
It´s an odd function f(-x)= -f(x) so symmetric on point (0,0)
lim y x==<+- infinity = +- infinity
y´=5x^4-12x^2+5 =0 x^2=z
z=((12+-sqrt(144-100)/10let´stake approximations
x=+-1.36 and x= +-0.73
y´´= 20x^3-24x^2 (calculate its value for each x but it is easier to
study sign of y
++++++-1,36-------(-0.73)++++++(0.73)---------(1.36)++++++
Max min Max min


3)do it yourself
4 )1/cos@=4cos@ so cos @ =+-1/2
taking0<@<2pi @= pi/3,5pi/3,2pi/3 and 4pi/3
5) if (xo,yo) is a point of E and (xo,y1) a point of C
the tangents are
(x*xo)/a^2+y*yo)/b^2=1 and x*xo+y*y1=a^2
put y=0 for the intercept
For E tangent x= a^2/xo and for C tangent x= a^2/xo (the same)
Thats enough