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Problem 48 Medium Difficulty

If $ \displaystyle \int^8_2 f(x) \, dx = 7.3 $ and $ \displaystyle \int^4_2 f(x) \, dx = 5.9 $, find $ \displaystyle \int^8_4 f(x) \, dx $.

Answer

$\int_{4}^{8} f(x) d x=1.4$

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Video Transcript

given that we have the whole area of the integral from 2 to 8, and that is 7.3. Let's go ahead and just sketch that out here. I just drew an arbitrary graph, so this is not exactly what it looks like. Of course I'm just using this as an example. But that tells us that the area here under the curve would all be 7.3. And then we're told from 2 to 4 that the area is 5.9. So then this area here would be 5.9. The question is, what is the area than from 4 to 8. So this area here, what would that evaluate too? And to do that, we could simply subtract the total area 7.3 minus 7.3 and then minus our 5.9. And that will end up giving us just go ahead and do that out. Six. That was before and 1.4. Therefore, the missing area here must be 1.4. Because if we were to add that area 1.4 plus 5.9, that would then get us back to 7.3 as well. So that is what the missing area for the integral from 4 to 8 would have to be here.