# Find the area of the shaded region?

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#x=y# and #x=1/y^2#

Please see below.

When we first learn to find areas by integration, we take representative rectangles vertically.

The rectangles have base

For this new problem, we could use two such intergrals (See the answer by Jim S), but it is very valuable to learn to turn our thinking

We will take representative rectangles horiontally.

The rectangles have height

Notice the duality

The phrase "from the smallest

The phrase "from the smallest

Here is a picture of the region with a small rectangle indicated:

The area is

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Area of the shaded region is

One of many ways the area of the shaded region can be expressed could be as the area of the triangle

Let

The area of the small triangle

#color(green)(Ω_2)=# #1/2*1*1=1/2m^2#

The area of

As a result, the shaded area will be

#Ω_1# #+color(green)(Ω_2)# #=1/2+1/2=1m^2#

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To find the area of the shaded region, you need to subtract the area of the unshaded region from the area of the larger shape (often a rectangle or circle). To do this, you'll first need to identify the geometric shapes involved and determine their dimensions. Then, you can use the appropriate formula to calculate the area of each shape, and finally, subtract the area of the unshaded region from the total area to find the area of the shaded region.

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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.

- How do you find the area bounded by #y=4-x^2#, the x and y axis, and x=1?
- Find the dimensions of the rectangle of maximum area whose perimeter is 16 cm ?
- How do you find the area between #y=-3/8x(x-8), y=10-1/2x, x=2, x=8#?
- Write a definite integral that yields the area of the region. (Do not evaluate the integral.)?
- How do you find the area between #y^2=-4(x-1)# and #y^2=-2(x-2)#?

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