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*}

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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 \[ \iint_A xy \, \de{x} \de{y}. \]

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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 \[ I = \int_0^a \int_0^x f(y) \, de{y} \de{x} = \int_0^a (a-y) f(y) \, \de{y}. \]

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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.

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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.

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