Math Problem Statement

Determine the limits of integration for the shaded rectangular region in the graph and set up the double integral.

Solution

The problem involves determining the limits of integration for the shaded rectangular region in the graph.

Based on the graph:

  1. The region spans from x=1x = 1 to x=2x = 2.
  2. It spans from y=0y = 0 to y=1y = 1.

Thus, we can fill in the values as follows:

  • a=1a = 1
  • b=2b = 2
  • c=0c = 0
  • d=1d = 1

The double integral over this region, in terms of xx and yy, can then be expressed as: 1201dydx\int_{1}^{2} \int_{0}^{1} \, dy \, dx

Would you like more details on setting up or evaluating this integral?


Related Questions:

  1. How do you determine the limits of integration for other shapes, such as circles or triangles?
  2. What does the result of the integral represent in terms of area or volume?
  3. How would this integral change if the region were rotated or reflected?
  4. Can the order of integration be reversed in this integral, and if so, how?
  5. How would the integral be set up if the function inside were non-constant?

Tip: When setting up double integrals, always check if the limits align with the bounds shown in the region graph.

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Math Problem Analysis

Mathematical Concepts

Calculus
Double Integrals
Limits of Integration

Formulas

Double Integral \( \int_a^b \int_c^d f(x,y) \,dy\,dx \)

Theorems

Fubini's Theorem

Suitable Grade Level

College Calculus (Calculus II or III)