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Calculate the area between the two curves below?
- MathguyLv 52 months ago
Probably easiest to Use sideways TE of Area .... that is integrate horizontally
∫ [ ( x value... on the right ) - ( x value... on the left ) ] dy
limits of integ. go from lowest to highest y values in common...
so rt side x + y = 2 thus x rt = 2 - y ... similar for left side
∫ [ ( 2 - y ) - ( √ y ) ] dy limits go from y = 0 to y = 1
[ 2y - (y^2) / 2 - (2/3) y^(3/2) ] from 0 to 1
You get 2 - (1/2) - (2/3) = 5 / 6
or split into two integrals using vertical Typical Elements [ TE ] of area ..... vertical rectangles...
- llafferLv 72 months ago
I would break this up into two sections.
First, you want the area under the curve of y = x² from x = 0 to 1, then you want the area under the curve of x + y = 2 (or y = -x + 2) between x = 1 and 2.
The second one is easy as it's a triangle with a known base (1) and height (1), so the area of that is:
A = bh/2
A = 1(1)/2
A = 1/2 unit²
To get the area under the curve of the first section we need to integrate (anti-derivative) of the function.
To do that we can add one to the exponent and then divide the original coefficient by the new exponent to get the new coefficient. There is normally a constant term here but since we are subtracting one integral from another the constant terms cancel out so we can ignore them.
y = x²
∫y = (1/3)x³
Now we solve this for x = 1 and subtract it from the value when x = 0:
(1/3)(1)³ - (1/3)(0)³
(1/3)(1) - (1/3)(0)
1/3 - 0
If we do the same thing to the triangle we will get the same area that we got using the area of a triangle equation, in case you wanted to give that a try for yourself, but I won't do that, here.
Finally, add the partial areas to get the total area:
1/2 + 1/3
3/6 + 2/6