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# Integrating sums of functions

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.A (LO)
,
FUN‑6.A.1 (EK)
,
FUN‑6.A.2 (EK)
If you know the integrals of two functions, what is the integral of their sum?

## Want to join the conversation?

• Im still a bit confused about the "Integrals" notation, ¿ what do we mean with dx? Im thinking it refers to (DeltaX) from the sum, but im not sure.
• When Δx becomes infinitesimally small, we call it dx, or the differential.
• Would it be correct to say that the definite integral of a sum is the sum of the definite integrals is a result of the limit property lim x->a f(x) + g(x) is the same as lim x->a f(x) + lim x->a g(x)? I'm not completely sure if dealing with the limit of the areas of geometric shapes with infinitely small widths follows the same rules as taking regular limits and if my intuition regarding this issue is correct.
• It is a combination of the limit addition property AND the sum (sigma notation) addition property (which is basically just the commutative property for addition). Just think of the definite integral as the Riemman sums. You add f(x_1) + g(x_1) + f(x_2) + g(x_2) + .... But you can use the commutative property and rewrite it as f(x_1) + (f_x2) + ... + g(x_1) + g(x_2) + ... So you see how the sum of f(x) + g(x) is exactly the sum of f(x) + the sum of g(x). Also the delta_x can be factored out of the original sum, and redistributed to the two resulting sums, so it does not affect out commutative property. Finally you end up with a limit of two sums, which can be split into two sums of limits.
• So is he taking the graph of y=g(x) and putting it on top of y=f(x)?

I'm visualizing him taking the graph of y=g(x), moving it, and when it's on top of the y=f(x), having the x-axis sort of shift from a straight line to a curve accommodating into the shape of y=f(x), and then the top of y=g(x) would change accordingly.

Is this an accurate description?
• The main take-away of this video, though it is not explicitly stated, is that the integral of the sum of two functions is equal to the sum of the integrals of each function, that is:
∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx.
Now since addition is commutative, the order of f(x) and g(x) does not matter.
Now, what Sal was doing was graphically showing this. The procedure he was using to illustrate goes something like this:
At x=a, add the heights f(a) and g(a) to get the combined height and graph it.
Now pick another point, x_1, such that a<x_1<b and add f(x_1) and g(x_1) to get the combined height and graph that.
Continue, for as many additional points as you care to, until you reach b, now add f(b) and g(b) to get the combined height and graph that.
Now connect the points you just graphed and that is the graph of f(x) + g(x) and the integral of that area ∫(f(x) + g(x))dx is equal to the sum of the areas of each of the functions f and g, that is:
∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx.
Hope that helped.
• Isn't this the similar idea as the sum rule for differentiaition?
• Correct - they are both linear operators. That is, integration and differentiation are linear operations.
• Good video. I have a question. What happens when we integrate (x+2). We can either do \int x+2 dx, or \int x dx + \int 2 dx. When x =2 first gives 18 and second gives 16. I've had something similar in an exercise i was doing about differential equations. I used the second method to solve it and the book used the first. We came up with different answers (after also substituting the original conditions), so i'm assuming that i'm wrong. I'm confused here. If someone can help with this, i'd appreciate it. Thanks in advance.
• What's the difference between indefinite and definite integrals?
• An indefinite integral of a function, also called an antiderivative of the function, is another function whose derivative is the original function. For example, suppose an antiderivative of 𝑓 is 𝐹. Then, the following equation is satisfied:
𝐹' = 𝑓
So indefinite integration is the reverse process of differentiation. Instead of going from the original function to its derivative, you're going from the derivative to an original function!

A definite integral WRT (with respect to) 𝑥 of a function of 𝑥 is the signed area bounded by the curve and the 𝑥-axis over some interval. For example, the definite integral WRT 𝑥 from 0 to 1 of the function 𝑓(𝑥) = 𝑥 is the area of the region bounded by the line from 𝑥 = 0 to 𝑥 = 1 and the 𝑥-axis. This region is a right triangle, and its area can be computed easily without any calculus (1/2). On the other hand, if we had a function like 𝑔(𝑥) = -𝑥, and we integrate WRT 𝑥 from 0 to 1, we are looking at an area underneath the 𝑥-axis. We define area underneath the 𝑥-axis to be negative area. This is what is meant by the term signed area: area above the 𝑥-axis is defined as positive and area under the 𝑥-axis is defined as negative.

So this may make you wonder why these two seemingly unrelated concepts both fall under the same name of "integration". In fact, there is a very useful and beautiful connection between these two concepts! This connection is called the Fundamental Theorem of Calculus. It states that the definite integral of the function 𝑓 WRT 𝑥 from 𝑥 = 𝑎 to 𝑥 = 𝑏 is equal to:
𝐹(𝑏) – 𝐹(𝑎)
Where 𝐹 is an antiderivative of 𝑓. I suggest you look into all of this on KA! Comment if you have questions!
• Are we basically adding two definite integrals?
• would the graph of f(x) + g(x) be the same as the graph of g(x) + f(x)?
(1 vote)
• Rules of addition/subtraction of function are the same as the addition of numbers: the Commutative Property holds.