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That's a good answer.I'm still confused about why we take the area in integration while the derivative just takes the slope of points.Actually why do we need to consider an area under the curve as said here:Integration when we only need to consider the function or more precisley the line or the curve that produced the function but not the points below that doesn't obey with the function.
I tried to look at the example over here:Math is Fun it says that when you integrate 2x you get x^2.It's right if you just look it based on theorems.Does x^2 at 3 means adding 2x '3' times in integration?But if you go this way you wouldn't get x^2 at x=3 which is 9 equal to 2*3=6.
If integration is adding I tried to add upto the three values at x=1,x=2 and x=3 but then the value becomes 12(2+4+6) using the equation 2x.How the value 0f 9 really came while integrating 2x at x=3 as shown in the example of filling a tank with water.Could anyone help me to get what really integration is other than saying it's just adding slices together to get a full piece.
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