text: # Basic double integrals
  
  ## 1. Simple integrals
  
  Evaluate \begin{gather*} \int_0^a \int_0^b xy (x^2 - y^2) \, \de{x} \de{y}, \\ \int_0^a \int_0^b xy \cos*x^2 y + y) \, \de{x} \de{y}, \end{gather*}
  
  Spoiler
  
  ## 2. Region integrals
  
  Let $A$ be the region of the points in the plane given by $x + y \geq 2$ and $x^2 + y^2 \leq 4$. Calculate $\iint_A xy \, \de{x} \de{y}.$
  
  Spoiler
  
  ## 3. Reversing the integration order
  
  Evaluate $\iint_R \sin(x + y \de{x} \de{y},$
  
  over the region $R$ bounded by the lines $y = x$, $y = 0$, and $x = a$ (where $a > 0$). Show that the result is the same if the order of integration is reversed. Next, show that $I = \int_0^a \int_0^x f(y) \, de{y} \de{x} = \int_0^a (a-y) f(y) \, \de{y}.$
  
  Spoiler
  
  ## 4. Reversing the integration order
  
  (a) Evaluate $\iint_R y \, \de{x} \de{y}$
  
  over the region $R$ bounded by the lines $y = x$, $y = 2 - x$, and $y = 0$. Write down the integrals which must be evaluated if the order of integration is reversed.
  
  (b) Change the order of integration in the repeated integral $\int_0^1 \int_x^{2-x} \frac{x}{y} \de{y} \de{x},$
  
  and evaluate the result.
  
  Spoiler
  
  ## 5. An integral with multiple bounding curves
  
  Sketch the region $R$ which is in the positive quadrant and is bounded by the curves $xy = 2, \quad y = \frac{x^2}{4}, \quad y = 4.$
  
  By integrating first with respect to $x$ and then $y$, find the area $R$. Check that the result is the same if you reverse the order of integration.
  
  Spoiler
explanation: The document contains multiple sections, each presenting a math problem involving double integrals. Each section includes a problem statement and a placeholder for the solution ("Spoiler"). The problems cover various aspects of double integrals, such as simple integrals, region integrals, reversing the order of integration, and integrals with multiple bounding curves.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Basic double integrals
  
  ## 1. Simple integrals
  
  Evaluate \begin{gather*} \int_0^a \int_0^b xy (x^2 - y^2) \, \de{x} \de{y}, \\ \int_0^a \int_0^b xy \cos*x^2 y + y) \, \de{x} \de{y}, \end{gather*}
  
  Spoiler
  
  ## 2. Region integrals
  
  Let $A$ be the region of the points in the plane given by $x + y \geq 2$ and $x^2 + y^2 \leq 4$. Calculate $\iint_A xy \, \de{x} \de{y}.$
  
  Spoiler
  
  ## 3. Reversing the integration order
  
  Evaluate $\iint_R \sin(x + y \de{x} \de{y},$
  
  over the region $R$ bounded by the lines $y = x$, $y = 0$, and $x = a$ (where $a > 0$). Show that the result is the same if the order of integration is reversed. Next, show that $I = \int_0^a \int_0^x f(y) \, de{y} \de{x} = \int_0^a (a-y) f(y) \, \de{y}.$
  
  Spoiler
  
  ## 4. Reversing the integration order
  
  (a) Evaluate $\iint_R y \, \de{x} \de{y}$
  
  over the region $R$ bounded by the lines $y = x$, $y = 2 - x$, and $y = 0$. Write down the integrals which must be evaluated if the order of integration is reversed.
  
  (b) Change the order of integration in the repeated integral $\int_0^1 \int_x^{2-x} \frac{x}{y} \de{y} \de{x},$
  
  and evaluate the result.
  
  Spoiler
  
  ## 5. An integral with multiple bounding curves
  
  Sketch the region $R$ which is in the positive quadrant and is bounded by the curves $xy = 2, \quad y = \frac{x^2}{4}, \quad y = 4.$
  
  By integrating first with respect to $x$ and then $y$, find the area $R$. Check that the result is the same if you reverse the order of integration.
  
  Spoiler
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains multiple sections, each presenting a math problem involving double integrals. Each section includes a problem statement and a placeholder for the solution ("Spoiler"). The problems cover various aspects of double integrals, such as simple integrals, region integrals, reversing the order of integration, and integrals with multiple bounding curves.
  
  Label: multiple-question