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