2 ms·
It's my first thought too, but I never developed an intuitive understanding of why the integral of x^2 is (x^3)/3 (even though I know it is by heart).
by raincole 3y ago
It's my first thought too, but I never developed an intuitive understanding of why the integral of x^2 is (x^3)/3 (even though I know it is by heart).
- chongli 3y agoDo you have intuitive understanding of the derivative rules? For example: d/dx of x^3 = lim h->0 of ((x+h)^3 - x^3)/(h) = lim h->0 of (x^3 + 3hx^2 + 3xh^2 + h^3 - x^3)/(h) = lim h->0 of 3x^2 + 3xh + h^2 = 3x^2 So then if we started with (x^3)/3 the 3s would cancel and we’d get x^2. This tells us that the antiderivative of x^2 is (x^3)/3. Or did you mean some other intuition? Such as why the fundamental theorem of calculus (that integration and differentiation are inverses of one another) is true?
- Izkata 3y agoGraphing y=x^0 -> y=x^1 and y=x^1 -> y=(1/2)(x^2) and doing math on the shapes (just simple squares and rectangles) makes those pretty intuitive. I just see it as an extension of that.