alright. i need to write a definite integral for the volume and evaluate the integral for the region bounded by y=x^2, y=0, and x=1 revolved about the a) x-axis, b) the y-axis, c) x=2, and d) y=-2.
HELP!
Lets do the x=2 axis.
dV= pi rh dr where r is radius, h is height. r= x-1 dr=dx h= y=x^2
dV= PI (x-1)x^2 dx
Then integrate from x=1 to 2
So the idea is to draw the pic, make an incremental dV in terms of r and h, change those to x,y variables.
What about rotating about y=6
dV= PI r h dr
r= 6-y y goes from 0 to 1^2
dr= dy
h= 1-sqrt(y)
check that picture
To find the volume of the region bounded by y = x^2, y = 0, and x = 1 when revolved about the x-axis, we can use the method of cylindrical shells.
First, let's find an expression for the volume of an infinitesimally thin cylindrical shell. The volume of a cylindrical shell can be represented as dV = 2πrh dx, where r is the radius of the shell and h is the height of the shell. In this case, r = x and h = y = x^2.
So, we have dV = 2π(x)(x^2) dx.
To find the total volume, we need to integrate this expression over the range of x from 0 to 1 (the bounds defined by y = 0 and x = 1).
∫(0 to 1) 2π(x)(x^2) dx
Evaluating this integral will give us the volume of the region when revolved about the x-axis.
Now, let's move on to the other cases:
b) To find the volume when revolved about the y-axis, we need to change our variables. In this case, the radius r becomes x, and the height h becomes the distance between the y-axis and the curve y = x^2, which is given by h = 1 - sqrt(y). The infinitesimal volume element is still given by dV = 2πrh dx. Integrate this expression over the range of y from 0 to 1^2, and that will give you the volume.
c) To find the volume when revolved about the line x = 2, we'll use a similar approach. Now, r becomes the distance between the line x = 2 and the curve y = x^2, which is given by r = 2 - x. The height h remains y = x^2. Again, the infinitesimal volume element is dV = 2πrh dx. Integrate this expression over the range of x from 1 to 2 to obtain the volume.
d) To find the volume when revolved about the line y = -2, we'll again change our variables. Now, r becomes the distance between the line y = -2 and the curve y = x^2, which is given by r = x^2 - (-2) = x^2 + 2. The height h remains x. The infinitesimal volume element is dV = 2πrh dx. Integrate this expression over the range of x from 0 to 1 to find the volume.
Remember to evaluate the definite integrals for each case to get the final volumes.