I'm asked to find the volume of the solid bounded by the curves x=1+y^2 and y=x-3, rotated about the y-axis.
I have a general idea of how to start the problem off, but I'm not sure how to find the interval that I need to work with...
First put each equation in the form x= .
There should be two intersections between the two equations, which will define the bounded area.
Okay, so when I write the integral, is it in terms of x or y?
It should be in terms of y, because you are rotating around the x axis. (That's why you have the form x = ).
*I meant y axis
Okay, that makes more sense now. Thanks so much!
To find the volume of the solid, you will need to integrate over a certain interval. In this case, since you are rotating the region about the y-axis, you need to determine the interval of y-values that corresponds to the region of interest.
To find this interval, you can set the two given curves equal to each other:
1+y^2 = x-3
Rearranging the equation, you get:
x = y^2 + 4
Now, you need to find the intersection points of the two curves. To do this, set x equal to each other:
1 + y^2 = y^2 + 4
Simplifying the equation, you get:
1 = 4
This equation is not true, which means there are no intersection points. Therefore, the interval will be from the lowest y-value to the highest y-value of the curves.
To find these values, you can find the y-values of the curves where they intersect with the y-axis.
For the curve x = 1 + y^2, substitute x = 0:
0 = 1 + y^2
Solving for y, you get:
y = ±√(-1)
Since y cannot be complex, there are no y-intercepts for this curve.
For the curve y = x - 3, substitute x = 0:
y = 0 - 3
y = -3
So the interval for y will be from -3 to the highest y-value of the curve x = 1 + y^2.
Now that you have determined the interval of integration, you can proceed with finding the volume of the solid by integrating the area of the cross-sections perpendicular to the y-axis using the method of cylindrical shells or washers.