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Area between curves

This video explains the method for finding the area between curves.
4.1
In this video, we are going to look at finding the area between curves.
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So we want to find area between two curves, and in the graph below, this black line is y equals x squared, and the red line is y equals x subtract x squared. We want to find this area that’s highlighted in pink, which is the area between the two curves. So the first step is to find this area in pink. Which x-coordinates does it lie between? So we can see it’s this and this one, which is where the two lines intersect. So we need to solve when do the two lines intersect. So we want y to be equal to y, which is when x squared is equal to x subtract x squared.
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We can rearrange this to find that 2x squared is equal to x, and that 2x squared subtract x is equal to zero. We can then take a factor of x out of both terms so that this becomes x lots of 2x subtract 1 is equal to 0. So we need to solve this equation to find the x-coordinates, where the two lines cross. So the first possibility is when x is equal to 0. And when x is equal to 0, y is equal to x squared, which is also 0. So 0, 0 is the first point where the two lines intersect. Alternatively, when 2x subtract 1 is equal to 0, this is when x is equal to 0.5.
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And the y-coordinate is when y is x squared, which is when y is 0.25. So the second point where these two lines intersect is 0.5, 0.25. So we found where the two lines intersect. So the next step is to start finding the area. So we first want to find the area which is underneath the red curve. So we want to find this whole area, which is underneath the red curve. And we’re going to find the area between x equals 0 and x equals 0.5 and bounded by the x-axis.
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So this is equivalent to finding the integral of the red curve, which is x subtract x squared, between the limits of x equals 0 and x equals 0.5. And this will get us this red area. When we integrate x, that’s x squared over 2. And when we integrate x squared, that’s x cubed over three. And we’ve got a definite integral between 0.5 and 0, so we first substitute 0.5 into there. And then we subtract the same thing with x equals 0 substituted in. And this will get us 1/12 is the definite integral. So 1/12 is the area underneath the red curve, and that’s been found using definite integrals.
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We then need to find the area that’s below the black curve instead.
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So that area, instead, we’ve just put on there that the area below the red curve is 1/12. So to find the area underneath the black curve, we would instead want to find this area, which can be found by finding the integral of x squared, which is the black curve between 0 and 0.5. If we integrate x squared, that gets us x cubed divided by 3, and we need to find out the definite integral between 0.5 and 0. So we first substitute 0.5 in to the formula, and then subtract the same thing with 0 substituted in. This gets 1/24 is the area under the black curve.
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So finally, we want the pink area, which is the area underneath the red curve subtract the area underneath the black curve. That area underneath the red curve was 1/12, and the area underneath the black curve is 1/24. So this gets us a final area of 1/24 is the answer.
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So in the comments below, you may want to discuss, can you find the area bounded by the x-axis and the line y is cosine of x between x equals 1 and x equals 2. And a hint, it might be helpful to draw a graph. Thank you for watching this video.

In this video, Dr Lisa Mott explains the method for finding the area between curves.

Share in the comments below:

Can you find the area bounded by the x-axis and the line y=cos(x) between x=1 and x=2?

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