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Expert-verified Found in: Page 699 ### Essential Calculus: Early Transcendentals

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

See the step by step solution

## Step 1: 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.

## Step 2: 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 $$x = 1$$.

## Step 3: 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 $$\int\limits_{x = 0}^1 {\int\limits_{y = x}^1 {f(x,y)dy} dx}$$. ### Want to see more solutions like these? 