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Q37E

Expert-verifiedFound in: Page 699

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**Sketch the region of integration and change the order of integration\(\int\limits_0^1 {\int\limits_0^y {f(x,y)dx} dy} \)**

We should know how to do graphical representation

Consider the double integral

\(I = \int\limits_0^1 {\int\limits_0^y {f(x,y)dx} dy} \)

The objective is to sketch the region of integration and change the order of integration.

Assume the integral where \(D\) is the domain of integration. The function \(f(x,y)\)is unknown.

The limits of integration for \(x\) varies from the line \(x = 0\) to line \(x = y\).

The limits of integration for \(y\) varies from the line \(y = 0\) to line \(y = 1\).

Therefore, the domain of integration is \(D = \left\{ {(x,y)|0 \le x \le y,0 \le y \le 1} \right\}\).

The region of integration bounded by the range of \(x\) and \(y\) that is line \(x = 0\), \(x = y\), \(y = 0\), and \(y = 1\) is shown in figure below.

Graph:

Take horizontal strip parallel to \(x\)-axis in the region covered in figure. It indicates that the strip moves from and \(y = 1\) with the end points lying on \(x = 0\) and \(x = 1\).

To evaluate the integral by change of order of integration. Consider a vertical strip which covers the region of integration.

The limits of integration for \(y\) varies from the line \(x = y\) to line \(y = 1\).

The limits of integration for \(x\) varies from the line \(x = 0\) to line \(x = 1\).

Therefore, the domain of integration is \(D = \left\{ {(x,y)|x \le y \le 1,0 \le x \le 1} \right\}\).

The region of integration bounded by the range of \(x\) and \(y\)that is the lines \(x = 0\), \(x = y\),\(y = 0\), and \(y = 1\) is shown in figure below.

Graph:

Take vertical strip parallel to y-axis in the region covered in figure. It indicates that the strip moves from and \(y = 1\) with the end points lying on \(x = 0\) and \(x = 1\)with the end points lying on \(x = y\)and \(y = 1\).

Thus, .

Hence, use of change of order of integration gives the value of integral as \(\int\limits_{x = 0}^1 {\int\limits_{y = x}^1 {f(x,y)dy} dx} \).

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