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