How do you find the area between #y=x, y=2-x, y=0#?

Answer 1

Let #A# be the area bounded by #y=x,y=2-x,# and #y=0#

#=>#
#A=int_0^1int_y^(2-y)dxdy=1#

First, a good thing to do would be sketch the graph.

So, we are looking for the area of that triangle.

Take note of the points of intersection.
Here we are dealing with #y=2-x# and #y=x#
So we set both equations equal and get

#x=2-x#

#<=># add #x# to both sides

#2x=2#

#<=># divide both sides by #2#

#x=1#

So, both functions intersect at #(1,1)#

Since we are only dealing with values greater than #0# we will have some #int_0^1fdy#

And since our area is bounded on the left by #y=x# and on the right by #y=2-x#, If we let #A# be our area we get

#A=int_0^1int_y^(2-y)dxdy#
#A=int_0^1int_y^(2-y)dxdy=int_0^1(2-y-y)dy=int2-2ydy#
#=2int1-ydy=2(y-y^2/2)]_0^1=2[(1-1/2)-(0-0/2)]=2[1/2]=1#

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Answer 2

To find the area between the curves (y = x), (y = 2 - x), and (y = 0), follow these steps:

  1. Identify the points of intersection between the curves (y = x) and (y = 2 - x). Setting (x = 2 - x), we solve for (x): [x = 2 - x] [2x = 2] [x = 1] When (x = 1), (y = x = 1). So, the curves intersect at the point (1, 1).

  2. Sketch the curves to visualize the area you need to find. Plotting (y = x), (y = 2 - x), and considering (y = 0) (the x-axis) helps in understanding how the area is bounded.

  3. Set up the integral to find the area. The area between these curves is essentially the area under (y = 2 - x) minus the area under (y = x), from (x = 0) to (x = 1) (the intersection point gives us the limit for the integration). The area (A) can be found by integrating the difference between the two functions over this interval: [A = \int_{0}^{1} [(2 - x) - x] , dx]

  4. Calculate the integral: [\begin{align*} A &= \int_{0}^{1} (2 - 2x) , dx \ &= \left[ 2x - x^2 \right]_0^1 \ &= (2(1) - 1^2) - (2(0) - 0^2) \ &= 2 - 1 \ &= 1 \end{align*}]

Therefore, the area between the curves (y = x), (y = 2 - x), and (y = 0) is 1 square unit.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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