I’m a relatively new knitter, and I’ve always been deeply paranoid about my knitting falling apart when I join two yarns, so I just tie a lot of knots instead of weaving in. However, this is pretty inefficient and can make the piece very unattractive. Can someone please explain to me how the piece doesn’t fall apart when the ends are woven in? I need some peace of mind before I do another project 
This might be of some help. If you leave a long enough tail and weave it in securely, there’s no chance of your knit items falling apart unless someone actually goes and pulls out the woven-in strand of yarn. Which would be a fairly tedious process and I don’t know why they’d do it in the first place.
I use the method tab posted because I am paranoid about that too! Haven’t had any problems.
Oh, and when joining two yarns, if they are felt-able you can felt them together! Amy has videos for ways to join yarn under the Basic Techniques…More section above.
I’ve made lots of sweaters and I’m sure I didn’t always weave in as securely as I could have. I’ve yet to have the slightest problem with anything unraveling or opening up. As long as the yarns are knitted/twisted/woven, even knotted sometimes, they’ll be fine. Sweaters generally don’t have any major stress points, if you think about it.
I will admit that I often do knot yarns when changing colours in knitting or crocheting. I do, however, leave long yarn tails and weave the tail in at least 2 directions for at least an inch or two {depending on the thickness of the yarn or thread}. When sewing a seam, I will also often use the seam for part of my weaving in. I have yet to have more than the very end appear in anything, and that’s usually my slippers which go through a lot more wear and tear than anything else I make. Even my DHs quiltghan which has been laundered over 20 times only has teeny {less than 1/4"} ends appearing. He singed it in our recent fire, so I guess I need to start working on a replacement ghan for him after Christmas. To make a long story short, unless the item is going to go through hell and back, tails don’t usually appear.