# 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
The text passage contains multiple sections, each with a different double integral problem to be evaluated, and some sections also include instructions to reverse the order of integration or to check the result with a different method.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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
user
The text passage contains multiple sections, each with a different double integral problem to be evaluated, and some sections also include instructions to reverse the order of integration or to check the result with a different method.

TYPE: MULTIPLE_QUESTIONS