Calculus: Volumes! Can you PLEASE help me with these?

Question

Find the Volume of the solid described. Give exact answers as reduced fractions in terms of pi.

1. The base is a circle with radius a. Find the volume if cross sections perpendicular are squares.

2. The base is the region enclosed by a circle of radius 7cm: find the volume if cross sections perpendicular to a fixed diameter are equilateral triangles.

3. Base is region bounded by y=1/x, y=0, x=1, x=4 and cross sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the xy-plane.

4. The base of a solid is bounded by x=y^2 and x=4. Find the volume if cross sections are:
a) perp. to -axis and are isosceles triangles with base in xy-plane and height is 1/2 of base.
b) perp. to x-axis and are ososceles right triangles with one =~ side in the xy-plane.
c) perp. to y-axis and are squares with one side in the xy-plane.
d) perp. to y-axis and are semicircles with diameter on xy-plane.

please please help me...i am pleading. you have no idea. help... :(

Answer 1

you need to do integral in 2nd degree for your equations.
if the shape is not a function (for example the circle) divide the problem to several intervals where it is, for example 1.
x^2+y^2 = a^2, Z=C
[-a,a] y = sqrt(a^2-x^2); area=SS[-a,a][0,C]sqrt(a^2-x^2)
[-a,a] y = -sqrt(a^2-x^2); area=SS[-a,a][0,C]-sqrt(a^2-x^2)
sum the two areas and you have your answer.