I’m in CalcII now and for the life of me cannot wrap my head around integrals. Now we are using things like u-Substitution methods and solving the areas between 2 curves. I can understand how the equations work, but not why because I still cannot picture what an integral is or why it’s important.
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If you have a function that’s 1 everywhere (f(x)=1), then when you’re at 0 on the X axis, Y is already at 1, but the area is 0, because the shape you have under the graph has no width yet. when you’re at 1 on the x axis, the area under the part you have checked is 1 unit, because it draws a horizontal line from (0, 1) to (1, 1). Move to 2 on the x axis and the area will be 2 because you’re increasing it at a constant rate. The area under the function between x=0 and x=1 is a 1×1 square and you add a square to it in every step. The integral of f(x)=1 is f(x)=1x, because the area under the line increases at a linear rate, and that rate is 1.
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If you start with f(x)=2, then for every step on X, the area increases by 2, because instead of 1×1 squares, you get 1×2 rectangles. The integral of f(x)=2 is f(x)=2x, because for every step on the X axis, the area will be twice the value of X
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Now take a function whose Y value increases by 1 for every X value, like f(x)=2x. For every step on X, Y goes up by 2.
At the first step, the area will be 1, because at X=0, Y was also 0 and got up to 2 as a linear function, cutting the 1×2 rectangle into 2 triangles.
When you get to 2, you add the same triangle under your graph, but also a full rectangle that’s lined up with the previous triangle.
Your area was 1, now you added 3 to it, so it’s 4.
Next step, you start with 4, add a triangle and 2 rectangles to fill the area under the new triangle, which are 5 area units in total.
Add that new 5 to the existing 4 and you get 9.
If you look at the areas we got, they are 1, 4 and 9, which are 1², 2² and 3². The integral of f(x)=2x is f(x)=x²
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Finding the derivative function is the opposite of this. You look for how fast the function’s Y value “grows” at any point. The constant functions (f(x)=2) don’t change, don’t grow, so their derivative is f(x)=0. The linear function grows at a steady rate, the steeper it is, the highest the rate of that growth is. The derivative of f(x)=ax is f(x)=a, where ‘a’ is a constant number. The steepness, so the rate of growth of quadratic functions like x² grows at a steady pace, as we’ve seen in the integral example, so the derivative of x² is 2x
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