Calculus l - Integration

Integration! 

We can use common geometric shapes to approximate the area between a curve and the x-axis over some interval. Rectangles are often used to approximate area because they fit better side by side, have individual dimensions that are easier to alter (height and width), easier to measure over a grid, and have the potential to not under/overestimate as opposed to other shapes. A Riemann sum is a finite sum of individual rectangle areas (which ultimately gives us an estimate of the total area).  

Left and right Riemann sums:  

The number of sub-divisions we create within an interval will tell us how many rectangles we can fit in that interval. These subdivisions, however, don’t all have to be equal. The distance between one x-value and its consecutive x-value is a width. The height of the rectangle that forms with this width can vary. We can use either the upper left or right corner of each rectangle as that rectangle’s height (but we must stay consistent with whichever corner we pick for every rectangle). The height would be determined by plugging in the x-value right below or above that corner into the function. Then we just sum up these individual areas to get an approximation of the total area in that interval. But doing so would cause over or under estimations. Both options have their flaws, after all, they are simply approximations. 

If a function is increasing, then the maximum length will always occur on the right corner side of each interval (giving us an overestimate), and the minimum length will always occur on the left corner side of each interval (giving us an underestimate). 

                                                                                                                                    

If a function is decreasing, then the maximum length will always occur on the left corner side of each interval (giving us an overestimate), and the minimum length will always occur on the right corner side of each interval (giving us an underestimate). 

                                                     

We can also see that the more rectangles we use, the better the approximation becomes. Having more rectangles means that each rectangle would need to have an increasingly small width to squeeze in more. The width would become so small that it’ll start to look like a single x-value. Having an increasingly small width means that the heights from either corner would also become closer and closer… it’ll almost look like a single straight line that stops at a single y-value. You can picture it as a ton of straight lines that fill up the area and are perfectly the height of the function at each individual x-value and there are infinitely many x-values we can fit in an interval. 

                           

This is where a limit comes, we say that the smaller the better, so we want the subdivisions to be as small as they can get, in return, make the width as small as possible. This means that even the max width must approach 0. But there’s a better, shorter way to write this using “∫” (integral symbol). Below, left of the equal sign is known as the definite integral.  

                                              

So, what does it mean to integrate something? 

Integration: The process of finding the antiderivative given its derivative is called integration or anti-differentiation. It’s commonly used to find the area under a curve. 

Integral = antiderivative = original function = F(x) 

∫ f (x) dx = F(x) + C 

where "f (x)” is the derivative of the original function or F‘(x) = f (x), “C” is a constant, “dx” means to integrate with respect to x.  

In terms of area, 

“f (x)” is the height of a point on the curve (the y-value of an x-value), and “dx” is the corresponding width (horizontal change in distance from one x-value to another x-value). 

 

Fundamental Theorems of Calculus (F.T.O.C): 

Pt. 1: There is an opposite process of differentiation known as integration. The antiderivative is the function before differentiation.  

When the lower bound is a constant, and the upper bound has a variable, if you take the derivative of the integral with respect to that variable, it will just give you the function of that variable multiplied by the derivative of the upper bound.  

                                         

If the variable is in the lower bound, as opposed to the upper bound, then you could multiply the integral by –1 and that will switch the upper and lower bounds: 

                                

If there are variables in both the lower and upper bounds, you can separate the integral into two integrals: 

   

Pt. 2: 

Definite integrals: an integral that has a lower and upper bound. This can be used to find the area within the interval (the bounds create the interval). 

If a function, f(x), is continuous over the interval (from lower to upper bound), and if F(x) is an antiderivative of f(x) on that interval, then: 

                                                        

 

 

 

Trig Antiderivatives: 

 

Other Antiderivatives rules and formulas that are good to have memorized: 

 

U-substitution method: 

Reverses the chain-rule in differentiation. It is also a good thing to use when you are dealing with functions being multiplied or divided. When you see that there is a function inside another function somewhere in the integral, you can set the inner function to equal “u” and find the derivative of u. Finding the derivative of u (du) with respect to the variable (dx) allows you to rewrite it in terms of “dx”. Doing that will allow you to replace the “dx” in the integral with the new value “dx” equals (du/(actual derivative of u)). This will allow you to cancel a part of the integral and it should simplify down to a single function of “u”. Lastly, just find the anti-derivative of the function that remains with respect to “u” and replace “u” with its actual value. Make sure to always add the +C !