4.7.4 Practice set: multiple integrals

Double integrals

Double integrals (lesson 4.7.1)

0 of 7 correct

1.
Estimate $\iint x^2 y\,dA$ over $[0, 2] \times [0, 1]$ with 2 cells, $[0, 1] \times [0, 1]$ and $[1, 2] \times [0, 1]$ , and midpoint heights
2.
Find $\int_0^1 \int_0^3 (x + y)\,dy\,dx$
3.
Find $\iint e^x \sin y\,dA$ over $[0, 1] \times [0, \pi]$
4.
Find $\iint_T y\,dA$ over the triangle $0 \le y \le x \le 1$
5.
$R$ lies between $y = x^2$ and $y = 1$ for $-1 \le x \le 1$ . Find $\iint_R y\,dA$
6.
Write $\int_0^1 \int_{x^2}^{1} f\,dy\,dx$ in the order $\int_0^1 \int_0^{?} f\,dx\,dy$ . What is the upper limit, in terms of $y$ ?
7.
Find $\int_0^1 \int_x^1 \sin(y^2)\,dy\,dx$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Change of variables

Change of variables and the Jacobian (lesson 4.7.2)

0 of 6 correct

1.
The map $x = u + 2v$ , $y = 3u + 4v$ multiplies areas by what factor?
2.
The map $x = 2u$ , $y = u + v$ sends $[0, 2] \times [0, 3]$ to a parallelogram. What is its area?
3.
$R$ is the region where $1 \le x + 2y \le 3$ and $0 \le x - y \le 3$ . Find its area
4.
For the same region, find $\iint_R (x + 2y)\,dA$
5.
With $x = u^2$ , $y = v$ for $u > 0$ , $\iint f\,dA = \iint f(u^2, v) \cdot k\,du\,dv$ . What is $k$ ?
6.
With $u$ and $v$ given in terms of $x$ and $y$ , a student finds $\det = 4$ and multiplies by 4. What factor should multiply $du\,dv$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Polar coordinates

Polar coordinates and the Gaussian integral (lesson 4.7.3)

0 of 7 correct

1.
Find the angle $\theta$ , between $-\pi$ and $\pi$ , of the point $(-2, 0)$ (type pi for $\pi$ )
2.
Find the area of the wedge $0 \le r \le 2$ , $0 \le \theta \le \frac{\pi}{3}$
3.
Find $\iint \frac{1}{x^2 + y^2}\,dA$ over the ring $1 \le r \le 3$
4.
Find $\iint e^{-(x^2 + y^2)}\,dA$ over the disc of radius 2, exactly
5.
The quarter plane $x, y \ge 0$ gives $\left(\int_0^\infty e^{-x^2}\,dx\right)^2 = \frac{\pi}{4}$ . Find $\int_0^\infty e^{-x^2}\,dx$
6.
Find $\int_{-\infty}^{\infty} e^{-3x^2}\,dx$ by squaring it (type sqrt and pi)
7.
Find the volume of the cone $z = 1 - \sqrt{x^2 + y^2}$ over the unit disc

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

A midpoint rule over a region

Write region_midpoint(f, a, b, lower, upper, n=200). It estimates $\iint_R f(x, y)\,dA$ over the region $a \le x \le b$ , $\text{lower}(x) \le y \le \text{upper}(x)$ , as in lesson 4.7.1.

Cut $[a, b]$ into n strips of width $\Delta x = \frac{b - a}{n}$ , and take the midpoint $x$ of each.
At that $x$ , cut the slice from lower(x) to upper(x) into n cells of height $\Delta y$ , and add $f$ at each cell's midpoint times $\Delta x\,\Delta y$ .
Raise ValueError if n is less than 1.

The tests try a rectangle, the region between $y = x^2$ and $y = x$ , the triangle with $e^{x^2}$ , the unit disc and the Gaussian of lesson 4.7.3. One checks that the error shrinks 4 times when n doubles, as a midpoint rule's should.

See the tests

import math

import numpy as np


def test_rectangle():
    """x² + 2y² over [0, 2] × [0, 1] gives about 4"""
    got = region_midpoint(lambda x, y: x**2 + 2 * y**2, 0, 2, lambda x: 0 * x, lambda x: 0 * x + 1)
    expect_close(got, 4.0, what="region_midpoint over the rectangle", rel=1e-4)


def test_one_cell():
    """With n = 1 the estimate is f at the middle, (1, 0.5), times the area 2: exactly 3"""
    got = region_midpoint(lambda x, y: x**2 + 2 * y**2, 0, 2, lambda x: 0 * x, lambda x: 0 * x + 1, n=1)
    expect_close(got, 3.0, what="region_midpoint with n = 1 (heights at midpoints)", rel=1e-12)


def test_not_symmetric():
    """x y² over [0, 2] × [0, 1] gives about 2/3, not 4/3: x and y keep their places"""
    got = region_midpoint(lambda x, y: x * y**2, 0, 2, lambda x: 0 * x, lambda x: 0 * x + 1)
    expect_close(got, 2 / 3, what="region_midpoint of x y²", rel=1e-4)


def test_between_curves():
    """xy between y = x² and y = x gives about 1/24"""
    got = region_midpoint(lambda x, y: x * y, 0, 1, lambda x: x**2, lambda x: x)
    expect_close(got, 1 / 24, what="region_midpoint of xy between x² and x", rel=1e-4)


def test_triangle():
    """e^(x²) over the triangle 0 ≤ y ≤ x ≤ 1 gives about (e − 1)/2 = 0.859141"""
    got = region_midpoint(lambda x, y: np.exp(x**2), 0, 1, lambda x: 0 * x, lambda x: x)
    expect_close(got, (math.e - 1) / 2, what="region_midpoint of e^(x²) over the triangle", rel=1e-4)


def test_disc():
    """The area of the unit disc, with y from −√(1 − x²) to √(1 − x²), is about π"""
    got = region_midpoint(lambda x, y: 0 * x + 1, -1, 1, lambda x: -np.sqrt(1 - x**2), lambda x: np.sqrt(1 - x**2))
    expect_close(got, math.pi, what="region_midpoint of 1 over the unit disc", rel=5e-4)


def test_gaussian():
    """e^−(x² + y²) over [−5, 5] × [−5, 5] gives π to 8 digits (lesson 4.7.3)"""
    got = region_midpoint(lambda x, y: np.exp(-x**2 - y**2), -5, 5, lambda x: 0 * x - 5, lambda x: 0 * x + 5)
    expect_close(got, math.pi, what="region_midpoint of the bell", rel=1e-8)


def test_second_order():
    """Doubling n from 50 to 100 cuts the error about 4 times"""
    f = lambda x, y: x**2 + 2 * y**2
    e50 = region_midpoint(f, 0, 2, lambda x: 0 * x, lambda x: 0 * x + 1, n=50) - 4
    e100 = region_midpoint(f, 0, 2, lambda x: 0 * x, lambda x: 0 * x + 1, n=100) - 4
    ratio = e50 / e100
    assert 3.5 < ratio < 4.5, f"expected the error to shrink about 4 times, it shrank {ratio:.3g} times: are the heights taken at the midpoints?"


def test_bad_n():
    """n = 0 raises ValueError"""
    try:
        region_midpoint(lambda x, y: 1.0, 0, 1, lambda x: 0.0, lambda x: 1.0, n=0)
    except ValueError:
        return
    raise AssertionError("expected ValueError for n = 0")

Double integrals

Change of variables

Polar coordinates

Coding exercise

Free further reading