Question

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

Short answer

We should know how to do graphical representation

Step-by-step solution

Given data:

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.

Graphical representation:

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 \(x = 1\).

Graphical representation:

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 \(\int\limits_{x = 0}^1 {\int\limits_{y = x}^1 {f(x,y)dy} dx} \).