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# Examples leveraging integration properties

Watch how some integral properties can actually be used.

## Want to join the conversation?

• at , Sal says that e^(x-2) is a shifted function of e^x. In the "definite integral of shifted function" video, we learned that f(x-c) is a shifted function. I understand that (e^x)-2 is a shifted function of e^x, but I don't understand how both (e^x)-2 and e^(x-2) can result in the same shifted function. (e^x)-2 and e^(x-2) have very different values. Is the answer that the function of x is as a power of e so that f(x-2) is e^(x-2)? If so, what would the area under (e^x)-2 equal? the sum of the areas under e^x and 2 between a and b, which would be an addition of 2 and not a shift? • The graph of e^(x-2) is the graph of e^x shifted right by 2. The graph of (e^x) -2 is the graph of e^x shifted down by 2. You are correct that they are totally different. I don't think he talks about the second one in this video.
So if you shift a graph to the right, the integral will be the same if you shift the limits of integration to the right. Thus in this example, the integral of e^(x-2) from x=2 to x=3 was the same as the integral of e^x from x=0 to x=1.
If you shift a graph up or down, the area under the curve will change by the area of a rectangle with width equal to the width of the interval of integration, and height equal to the height of the vertical shift. The integral of e^x from x=0 to x=1 was e - 1, so the integral of (e^x) -2 from x=0 to x=1 is (e - 1) + (1 * -2) = e - 3.
• Negative areas were touched on in the previous video dealing with switching the bounds. In the second function, Sal sketched the constant portion of the function above the x axis. Am I right in thinking the rest of the function would be the original graph flipped over under the x axis?   