Undergrad Integral power rule explanation

[YoungPhysicist]

I am new to the world of calculus and the first thing that I learned is how to calculate the area under the range of a polynomial function, like:

$$\int_1^3 x^2 \,dx$$

when I take the intergal of $x^2$, I get $\frac{x^3}{3}$due to the power rule,
but it doesn’t make sense to me,why would you need to divide the coefficient by the exponent and add the exponent by 1?Is there a simple explanation to why such rules exist?

ps:I may use some wrong terms,sorry for that.

 

[Svein]

Think of integration as the opposite of derivation.

 

[YoungPhysicist]

Think of integration as the opposite of derivation.

Sorry,as I mentioned above,I am new to calculus and this is the first thing that I learned.
While I know what derivatives are,I don’t know how to calculate them. So if the power rule’s explanation has something to do with derivatives,I may have to come back to this problem later on my journey on calculus.

 

[fresh_42]

You can also think of an easier example. You mentioned the area, which is not 100% correct, since it is the oriented area, i.e. areas above and below the x-axis have different orientations and thus different signs, but it serves the goal here. So if we have $f(x)=x$ then this area will be a triangle. Therefore you cannot calculate width $(x)$ times height $(f(x)=x)$, as this would be a rectangle. Triangles have half this area, so we must divide by two.

Same here for $f(x)=x^2$. A multiplication, i.e. area calculation width $(x)$ times height $(f(x)=x^2)$ would result in a rectangle. However, we don't have this entire area, not even half of it, since our curve is still below the halving diagonal. It's now merely a third of the area of the rectangle.

 

[Mark44]

when I take the intergal

There's nothing "inter" about it. The word is integral, and is related to the word integer.

Sorry,as I mentioned above,I am new to calculus and this is the first thing that I learned.
While I know what derivatives are,I don’t know how to calculate them. So if the power rule’s explanation has something to do with derivatives,I may have to come back to this problem later on my journey on calculus.

Yes, you should postpone this study until you learn how to calculate derivatives.