I'm currently working on an assignment that has some "fairly" abstract ideas that need to be utilized, and we never went over these specific scenarios in class...
I'm only looking for direction on how to go about solving...
#1.) There is a line going through the origin that divides the region bounded by the line y = x - x^2 and the x-axis into two sections of equal area. Find the slope of this line.
Now, I found the area of the region to be 1/6 and therefore, half of it should be equal to 1/12. I don't know how to determine where to cut the interval off into two pieces so that I can solve for the slope of the line...
#2.) How can I show that two regions have the same area without evaluating any integrals (the prof specifically told us that we needn't use any). The equations for the lines are:
y = x / (1 + x^4) from 0 to 2
y = 1 / (2(1 + x^2)) from 0 to 4
I have no idea where to begin, other than using integrals to evaluate them...
#3.) If R is the region bounded by y = sin (x^2) and the x-axis from x = 0 to x = pi^(1/2), let S be the solid of revolution if R is rotated about the y-axis. Find the Volume of S.
I used Cylindrical shells to find the volume and ended up getting 0 for my final answer. Anyone else ever had that happen? (My calculations have been checked...)
#4.) I need to find a function f that makes the following true:
The integral from 0 to pi of x(f(sinx))dx = the integral from 0 to pi of x(sinx)
I then need to evaluate the integral on the right side...any thoughts?
Again, looking for direction, not mooching answers...