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Question

Find the area of the shaded region webassign plot.

Answer “Find the area of the shaded region webassign plot”.

Area under the Curve

The area under the curve can be calculated as the method of integration with the particular limit of the function. The area is determined by the equation of the curve and their limits along the x-axis and the y-axis.

Area of Regions using Double Integrals:

The area R that is located situated on the surface that is x-y is calculated with using an integral:

A =

Type I Region

If this region is one of the Type I region, the solution is made easier since the integral mentioned above can be used in the form of an iterated integral.

A type I region is a region that can be expressed as the set:

$\begin{array}{c}R=\left\{\right(x,y):y\in [g\left(x\right),h\left(x\right)],x\in [a,b\left]\right\}\end{array}$

where $g\left(x\right),h\left(x\right)$ are continuous.

In this case:

Type II Regions:

When solving the problem of a double integral on the region R by using the technique of iterated integrals it is essential to look at the specifics of the integration region.  If the points belonging to the region R are between two continuous functions of the coordinates, then it can be possible that the double integral can be expressed as an iterated integral.

Furthermore, if the x-coordinates of the points within the area R are tied to two different functions$x=g\left(y\right)$ and $x=h\left(y\right)$, with $g\left(y\right)<h\left(y\right),\forall (x,y)\in R$ and the y-coordinates are contained in the interval $[c,d]$, this region is called as Type II region, and it can be expressed as:

R={(x,y): x ∈ [g(y), h(y)], y ∈ [a,b]}

For a Type II region, the double integral:

Can be expressed as:

Double Integrals and Areas:

If we are looking to determine the area of a certain zone R, it is possible to find it with double integrals, as:

Answer and Explanation:

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