Can you multiply integrals

So that's one term. The other one is notice, I said sum even though there's a negative sign here. I can pull that negative inside.

Then I can pull the constants outside. Pretty soon I'll be doing these two steps in a single step. Once you get used to all the properties, but for right now I'm separating them. So I want to pull this -4 out. So now I have each integral as just the integral of some power of x. So x to the 11 over 3. I add 1. Some cancellations happens. The 22 and the 11 cancel leaving a 2.

The 4 and the 2 cancel leaving a 2. Previous Unit Applications of the Derivative. Next Unit The Definite Integral. Norm Prokup. Thank you for watching the video. Select the second example from the drop down menu. This shows a line and the area under the curve from a to b in green.

Also shown is a second function, in red, which is a constant multiple c of the first function i. What do you notice about the areas values of the areas are shown in the top left corner of the graph? Drag the c slider, or type different values for c into the c input box. What do you notice? Try making c be This illustrates the constant multiple rule : In other words, if the integrand in a definite integral is multiplied by a constant, you can "pull the constant outside" the integral.

Select the third example. What do you notice about the areas? Move the a and b sliders and see if this relationship still holds. You should notice that the area under the red curve is the sum of the areas under the blue and green curves. This makes sense, since the Riemann sums are just made up of tall, thin rectangles and the height of the red rectangles is just the sum of the heights of the green and blue rectangles.

However there isn't a general approach on this problem. Add a comment. Active Oldest Votes. Stoof Stoof 4 4 silver badges 17 17 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. We can't multiply changing numbers, so we integrate. You'll hear a lot of talk about area -- area is just one way to visualize multiplication.

The key isn't the area, it's the idea of combining quantities into a new result. We can integrate "multiply" length and width to get plain old area, sure. But we can integrate speed and time to get distance, or length, width and height to get volume. When we want to use regular multiplication, but can't, we bring out the big guns and integrate.

Area is just a visualization technique , don't get too caught up in it. Now go learn calculus! That's my aha moment: integration is a "better multiplication" that works on things that change. Let's learn to see integrals in this light. We're evolving towards a general notion of "applying" one number to another, and the properties we apply repeated counting, scaling, flipping or rotating can vary.

Integration is another step along this path. Area is a nuanced topic. For today, let's see area as a visual representation of of multiplication :. With each count on a different axis, we can "apply them" 3 applied to 4 and get a result 12 square units. The properties of each input length and length were transferred to the result square units. Simple, right? Well, it gets tricky. We understand the graph is a representation of multiplication, and use the analogy as it serves us.

If everyone were blind and we had no diagrams, we could still multiply just fine. Area is just an interpretation. What's happening? Well, 4. We're taking 3 the value 4. We're so used to multiplication that we forget how well it works. We can break a number into units whole and partial , multiply each piece, and add up the results.