00:01
We are given the equations, three equations, given parabola, that parabola has equation y equals 4 x minus x squared, then we are given a line that has equation x equals minus 2, and then we are given another line, which is the equation y equals 4.
00:35
To find the area bounded by the above graphs.
00:50
So let us draw the graphs and see what is the area that we have to find.
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First, let us convert the parable equation, y equal to 4x minus x square.
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This can be written as y minus 4 equals minus x minus 2 whole square.
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So this parabola is inverted parabola, concave downwards from along the y -axis.
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So, and the lines are there.
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So we have to draw the graphs.
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This is the vertex, which is 24.
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This is 2 .0.
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This is 0.
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This is 4.
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This line is y equal to 4.
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This mix at this point.
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And this line is...
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X equal to minus 2.
02:42
This point is minus 2 0 and this point if we substitute minus 2 here, so minus 2 minus 2 is minus 4 and that is 16, there is minus 16, 4 minus 16, minus 16, 4 minus 16 is minus 12.
03:03
So this point is minus 2, minus 12.
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So this is the overall graph.
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So if we can color it with different colors.
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This is the area we have to find area.
03:37
We have to find.
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This is the parabola y equals 4x minus x square.
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This is the x axis.
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This is the y axis.
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This is the line x equal to minus 2.
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This is the line y equal to 4.
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And this is the required area which we have to find.
04:19
So how do we find this? first let us find this particular area.
04:25
First let us find this area the blue shaded area so how do you find this let us call this area blue and we have to find this which let us denote as area purple so what is area blue area blue is integration of y is 4x minus x square and here the y is positive so we have to take from 0 to 2.
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So from 0 to 2, 4x minus x square d x.
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This is area blue.
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So this if we compute, it will be x squared.
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So 2x squared minus x cubed by 3 to 0.
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So this will be equal to 2 to 48 minus 8 over 3.
05:55
So this is 2x square minus x -q over 3, range is 0 to 2, and this on computation, i'm putting the values, it becomes 8 minus 8 over 3, which is equal to 24 minus 8, 16 over 3.
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So we have found the blue area.
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This area we have found.
06:14
Now, let us find what is the area of, what is the area of this square, or rather this rectangle.
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I'll just put a different color here.
06:49
Let's just make it a little bit thick so it will be easier to understand.
06:57
So this green rectangle, what is the height? this is 2 and the length is 2, this is 2 and this is also 2, so this is 4.
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So the area of the green rectangle, area of the green rectangle equals 4 into 2, 4 multiplied by 2 or 4 times 2 which is equal to 8 units or 8 square units.
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What is the area of this portion? it will be 8 minus the area shaded in blue.
08:02
So let us say this area, area at the top...