# Find the area of the shaded region?

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

##### 2 Answers

Please see below.

#### Explanation:

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

Area of the shaded region is

#### Explanation:

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#