volumes solving for dx or dy

Question

The only problem I have with this is knowing when you are solving for dx or dy. For example, this question which says

find the volume of the solid created by rotating the region bounded by y = 2x-4, y = 0, and x = 3, about the line x = 4. This one is dy. Instead of y = 2x-4, the person solved in respect to x and got y+4/(2).

And then there is this one, similar to the above but about the line y = -3. This one is dx. I don't understand why. Help appreciated.

Answer 1

Sometimes it is easier to use horizontal strips than vertical (or the other way around). It also depends on the method you choose to find the volume, there is washer(disk) or shells. Shells you want parallel strips to the axis of rotation. Washer(disk) you want perpendicular strips to the axis of rotation. $x=f(y)$ think horizontal strip. $y=g(x)$ think vertical strip. In your problem you have you want to rotate the region bounded by $y=2x-4$, $y=0$, and $x=3$ about $x=4$. So your figure can be represented by a bunch of vertical strips. And vertical strips are parallel to vertical lines, the vertical line we talking about is the axis of rotation here. So I would do shells method here. $V=\int_2^3 2 \cdot \pi (4-x)(2x-4)dx $. Now if you want to look at your figure in terms of horizontal strips solve for $x$ given $y=2x-4$. This will mean we will use the washer(disk) method though. $x=\frac{y+4}{2}$. So $V=\int_0^2 \pi ((4-\frac{y+4}{2})^2-(4-3)^2) dy$

Answer 2

If the piece of the function rotates around the $x$-axis, you use $dx$

If it rotates around the $y$-axis, you use the inverse function and calculate the values of the function to get the limits of the integral.