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...
Cool question! This first thing I thought to do was find the equation of a tangent line to it and then find its antiderivative?? Just guessing. Trying to give you another option. I didn't try it out.
#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...
Maybe estimating rectangles and do a Reimann sum?
#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...)
Double check my calculations, but I think this is correct
Sorry don't have time to finish, hope this helps!