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Finding definite integrals using area formulas

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.A (LO)
,
FUN‑6.A.1 (EK)
Since definite integrals are the net area between a curve and the x-axis, we can sometimes use geometric area formulas to find definite integrals. See how it's done.

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  • starky tree style avatar for user Morgon
    So what if he combined the last two problems into one - sum1-6 f(x)dx? Is it just the sum of the areas of the triangle and semicircle, 6 + (-pi/2)?
    (3 votes)
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  • piceratops seed style avatar for user 2020mp9280
    I thought area under the curve meant that there couldn't be negative numbers
    (2 votes)
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  • winston default style avatar for user ‎
    This is very convenient to us with or knowledge so far, but what would we do if the equation was shifted down 1 unit (or less)? How would we do the first problem, using our current knowledge, without using trapezoidal sums or Riemann sums, since we're just dealing with a semi-circle?
    (2 votes)
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    • leafers seed style avatar for user Travis Bartholome
      Interesting question - I think you could still solve it geometrically, it's just a significantly tougher problem. Check out this resource: https://www.mathopenref.com/segmentarea.html

      You could compute the central angle given the radius of the circle and the height offset (however much you wanted to "shift the function down"). Computing the area of the circle segment that's greater than 0 gives the positive part of the area for x in [-6, -2]. The negative part could then be found by computing the area that's within the semicircle but outside of the positive segment, then subtracting that from the rectangle formed by y=0 and y=heightOffset to get the negative area for x in [-6, -2]. (Disclaimer: I'm just sort of mulling through this in my head, so I'm not sure if that's exactly right, but it seems like the correct approach.)

      That being said, the point of this problem is to be relatively convenient and provide a good way to grasp the concept at hand (integrals as areas under curves). The alternatives you listed are designed to solve more complex problems; not everything has to be done using the least available knowledge.
      (2 votes)
  • blobby green style avatar for user djlynes0
    Did sal forget to also calculate the other triangle in the interval [1,4]? .5(3*4)= 6 and .5(1*4)=2 then 2 + 6 = 8 I believe
    (1 vote)
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  • piceratops sapling style avatar for user Rayaan
    If we were to switch places with the numbers for each integral, for example, if we switched the 4 by the 6 and vice-versa in the 4th example shown in the video, would the area of the semi-circle be considered positive despite being below the x-axis? (I'm new to integration so I don't know much about all the properties yet, but I think I saw that somewhere, so I am not so sure if what I said is true)
    (0 votes)
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    • leaf green style avatar for user cossine
      Area is always positive.

      However any area underneath the x-axis is negative when perform the integration. If you remember the explanation Sal gave using rectangles to approximate an area you realise if f(x)<0 then area of the rectangle will come out as negative f(x) dx is less than 0.


      If switch the bounds of the integrand then the result will switch signs.

      Try integrating from some function f(x) from a to b will lead result of

      F(b)-F(a)

      while swapping the bounds gets you

      F (a)-F (b) = -( F (a) - F (b) ) which is opposite the above example
      (2 votes)

Video transcript

- [Instructor] We're told to find the following integrals, and we're given the graph of f right over here. So this first one is the definite integral from negative six to negative two of f of x dx. Pause this video and see if you can figure this one out from this graph. All right we're going from x equals negative six to x equals negative two, and the definite integral is going to be the area below our graph and above the x-axis. So it's going to be this area right over here. And how do we figure that out? Well this is a semicircle, and we know how to find the area of a circle if we know its radius. And this circle has radius two, has a radius of two. No matter what direction we go in from the center, it has a radius of two. And so the area of a circle is pi r squared. So it would be pi times our radius which is two squared, but this is a semicircle, so I'm gonna divide by two. It's only 1/2 the area of the full circle. So this is going to be four pi over two, which is equal to two pi. All right let's do another one. So here we have the definite integral from negative two to one of f of x dx. Pause the video and see if you can figure that out. All right let's do it together. So we're going from negative two to one, and so we have to be a little bit careful here. So the definite integral, you could view it as the area below the, below the function and above the x-axis. But here the function is below the x-axis. And so what we can do is, we can figure out this area, just knowing what we know about geometry, and then we have to realize that this is going to be a negative value for the definite integral because our function is below the x-axis. So what's the area here? Well there's a couple of ways to think about it. We could split it up into a few shapes. So you could just view it as a trapezoid or you can just split it up into a rectangle and two triangles. So if you split it up like this, this triangle right over here has an area of one times two times 1/2. So this has an area of one. This rectangle right over here has an area of two times one, so it has an area of two. And then this triangle right over here is the same area as the first one. It's going to have a base of one, a height of two, so it's one times two times 1/2. Remember the area of a triangle is 1/2 base times height. So it's one. So if you add up those areas, one plus two plus one is four, and so you might be tempted to say oh is this going to be equal to four? But remember our function is below the x-axis here, and so this is going to be a negative four. All right let's do another one. So now we're gonna go from one to four of f of x dx. So pause the video and see if you can figure that out. So we're gonna go from here to here, and so it's gonna be this area right over there. So how do we figure that out? Well it's just the formula for the area of a triangle, base times height times 1/2. So or you could say 1/2 times our base, which is a length of, see we have a base of three right over here, go from one to four, so 1/2 times three times our height, which is one, two, three, four, times four. Well this is just going to get us six. All right last but not least, if we are going from four to six of f of x dx. So that's going to be this area right over here, but we have to be careful. Our function is below the x-axis, so we'll figure out this area and then it's going to be negative. So this is a half of a circle of radius one. And so the area of a circle is pi times r squared, so it's pi times one squared. That would be the area if we went all the way around like that, but this is only half of the circle, so divided by two. And since this area is above the function and below the x-axis, it's going to be negative. So this is going to be equal to negative pi over two. And we are done.