Sketch the region D and express the double integral as an iterated integral with reversed order of integration.
Understand the Problem
The question asks for the sketch of a specific region D in the context of double integrals and requires expressing the given double integral with the order of integration reversed. This involves understanding the limits of integration and the geometric interpretation of the integrals.
Answer
The reversed double integral is: $$ \int_0^2 \int_{x/2}^1 f(x,y) \, dy \, dx + \int_2^3 \int_1^{3-x} f(x,y) \, dy \, dx $$
Answer for screen readers
The double integral expressed with reversed order of integration is:
$$ \int_0^2 \int_{x/2}^1 f(x,y) , dy , dx + \int_2^3 \int_1^{3-x} f(x,y) , dy , dx $$
Steps to Solve
- Analyze the Given Integrals
The integrals are given as:
$$ \int_0^1 \int_0^{2y} f(x,y) , dx , dy + \int_1^3 \int_0^{3-y} f(x,y) , dx , dy $$
- Determine the Limits for Region D
For the first integral, $0 \leq y \leq 1$ and $0 \leq x \leq 2y$.
For the second integral, $1 < y \leq 3$ and $0 \leq x \leq 3 - y$.
- Sketch the Region D
- For the first part, plot $y$ from 0 to 1 and within that, $x$ extends from 0 to $2y$, a line from (0,0) to (2,1).
- For the second part, plot $y$ from 1 to 3 and $x$ goes from 0 to $3 - y$, which is a line from (0,3) to (3,0).
Together these define a trapezoidal region.
- Find the New Limits of Integration
To reverse the order of integration, we now consider:
- From $y = 0$ to $y = 1$: for a given $x$, $y$ ranges from $x/2$ to 1 when $0 \leq x \leq 2$.
- From $y = 1$ to $y = 3$: for a given $x$, $y$ ranges from 1 to $3-x$ when $2 < x \leq 3$.
- Write the Reversed Integrals
Combining these, the reversed double integral becomes:
$$ \int_0^2 \int_{x/2}^1 f(x,y) , dy , dx + \int_2^3 \int_1^{3-x} f(x,y) , dy , dx $$
The double integral expressed with reversed order of integration is:
$$ \int_0^2 \int_{x/2}^1 f(x,y) , dy , dx + \int_2^3 \int_1^{3-x} f(x,y) , dy , dx $$
More Information
The region ( D ) is composed of two parts created by the two given iterated integrals. This involves understanding geometric interpretations and interpreting the limits corresponding to the specific curves and lines.
Tips
- Failing to correctly identify the boundaries when reversing the order of integration. Always ensure that the regions are accurately sketched and accounted for in the limits.
- Misunderstanding the relationship between ( x ) and ( y ) when inspecting the intersections of the lines.
AI-generated content may contain errors. Please verify critical information
