Original subtitles

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In this video we're talking about area under the curve versus area enclosed by the curve and the x axis

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and this is a less common application of definite integrals.

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Not all textbooks are going to ask you to make this distinction but if you run across a problem like

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this one this is probably what they're talking about.

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So area under the curve is by far the most common way to look at area.

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And when we talk about area under the curve it's also considered net area.

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So when you hear area under the curve.

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Think net area versus area enclosed by the curve and the x axis so enclosed would be a gross area.

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The same thing is a gross area and the difference is just with area under the curve or net area.

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You treat area above the x axis as positive an area below the x axis as negative.

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That's what you do when you take an integral of a function.

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Any area that's enclosed by the curve in the x axis that lies above the x axis is positive an area below

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the x axis is negative.

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So again because that's what we do when we integrate.

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That's by far the most common way to look at area.

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But sometimes you're going to hear this area enclosed by the curve and that means gross area and that's

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where retreat area above and below the x axis is positive.

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In other words we're taking absolute value of all of the area so we could right here.

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Area under the curve or net area we're going to be doing positive and negative area for area enclosed

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by the curb or gross area.

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We're going to be doing all positive area.

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So here's what that looks like.

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This is the information we've been given.

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We have this curve graphed in orange here.

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It's the curve f of x and notice at the interval for which it's defined as X equals negative 2 to X

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equals 6.

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We've also been given these three pieces of information here so we've been told that the integral of

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the function f of x so this function here in orange from negative to the positive one over that interval.

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So that's this interval right here negative to positive 1.

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So over this interval area is negative 2.8.

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And because this is the integral we're talking about area under the curve or net area where retreat

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area below the x axis as negative.

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So when this says negative 2.8 there acknowledging that this area from negative to positive one is all

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below the x axis and that the net area there is negative 2.8.

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Same thing here when we look at the integral of the function from 1 to 3 that means area under the curve

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or net area.

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So one to three is this interval.

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Right here from one to here three.

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So the area there is 1.2.

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It's positive because we can see that all of that area is above the x axis so it makes sense that they

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would indicate a positive area.

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Then this last integral here the integral from 1 to 6 is going to give us an area of negative 3.5.

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So before we're too quick to label this third area as negative 3.5.

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Notice that the interval here is 1 to 6.

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So that's the interval here from 1 all the way to 6 so this entire interval here we already know that

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that interval includes this positive area of 1.2.

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So if we go ahead and call this area right here a we know that taking the integral from to 5:59 looks

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at the net area or the area under the curve on this entire interval from one to six which means that

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it's going to treat this area between 1 and 3 as positive.

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And this area a between 3 and 6 as negative.

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So what we would want to say then is that positive 1.2 Plus a is going to be equal to the net total

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of negative three point five we would solve for a by subtracting 1.2 from both sides and we would get

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a is equal to negative four point seven.

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So we can say then that this area between 3 and six this entire area under the curve right here is negative

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four point seven.

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So in that area or area under the curve or the area you get when you take the integral when you take

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the integral if you get a negative result that means there's more area under the x axis than there is

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above the x axis.

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Similarly if you get a positive result that means there's more area above the x axis than below it.

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So now that we have our three areas if we want to find area under the curve over the entire interval

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negative to positive 6 the entire interval for the function we want to find.

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NET area.

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That means we take into account positive and negative values.

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So we would say that this is equal to negative 2.8 plus a positive 1.2 Plus a negative 4.7 or just minus

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4.7.

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And when we simplify here we get a negative 6.3.

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So that means the net area or the area under the curve is negative 6.3.

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That means that if we took the integral from negative to positive 6 we would get negative 6.3 because

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this net area is what we get when we take the integral and course it should make sense that we get a

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negative answer because if we look here at the graph we can see that we have more negative area more

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area under the x axis than we do above the x axis.

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If on the other hand we want to find area enclosed by the curve and the x axis or gross area we treat

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everything as positive.

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So we take the absolute value of each section of area that's enclosed by the curve and the x axis.

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So instead of negative 2.8 we say the absolute value of negative 2.8 Plus the absolute value of 1.2

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Plus the absolute value of negative four point seven.

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In other words we just treat everything as positive.

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So we end up with here is positive 2.8 plus a positive 1.2 plus a positive 4.7.

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And when we simplify that we get a positive eight point seven and that's the difference between area

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under the curve an area enclosed by the curve.