2.5.4 Practice set: integration

Riemann sums

Area and Riemann sums (lesson 2.5.1)

0 of 7 correct

1.
With 5 strips on $[2, 7]$ , what is $\Delta x$ ?
2.
Midpoint sum for $x^2$ on $[0, 2]$ with 2 strips
3.
A car drives at 80 km per hour for 1.5 hours. What is the area under its speed graph, in km?
4.
A left sum is off by 0.02 with 50 strips. About how far off is it with 500 strips?
5.
$\frac{1}{x}$ falls on $[1, 5]$ . With 4 strips, by how much does the left sum exceed the right sum?
6.
Left sum for $x^2$ on $[0, 2]$ with 4 strips
7.
$\int_{-2}^{2} x\,dx = 0$ . What is the total area between $y = x$ and the $x$ -axis from $-2$ to 2?

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.

Antiderivatives

Antiderivatives (lesson 2.5.2)

0 of 7 correct

1.
Find an antiderivative of $4x^3 - 6x + 2$ (leave out $+ C$ )
2.
Find an antiderivative of $\frac{1}{\sqrt{x}}$ (leave out $+ C$ )
3.
Find an antiderivative of $e^{-2x}$ (leave out $+ C$ )
4.
$F'(x) = 3x^2$ and $F(2) = 10$ . Find $F(x)$
5.
Find an antiderivative of $\frac{5}{x}$ for $x > 0$ (leave out $+ C$ )
6.
Find an antiderivative of $\frac{1}{x^3}$ (leave out $+ C$ )
7.
Find an antiderivative of $x(x + 2)$ (leave out $+ C$ )

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.

The fundamental theorem

The fundamental theorem of calculus (lesson 2.5.3)

0 of 7 correct

1.
Find $\int_0^2 3x^2\,dx$
2.
Find $\int_1^{e} \frac{2}{x}\,dx$
3.
Find $\int_0^1 e^{2x}\,dx$
4.
$A(x) = \int_0^x e^{t^2}\,dt$ . Find the slope $A'(1)$
5.
A pipe delivers $6t$ litres per minute. How many litres arrive between $t = 1$ and $t = 3$ minutes?
6.
Find $\int_4^1 \sqrt{x}\,dx$
7.
Find the total area between $y = x^2 - 1$ and the $x$ -axis from 0 to 2

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

Riemann sums with a choice of rule

Write riemann(f, a, b, n, rule="mid") that returns the Riemann sum of f on [a, b] with n strips. It takes heights at the "left" end, the "right" end or the "mid"point of each strip. Raise ValueError for any other rule.

f accepts NumPy arrays, so you can evaluate all the heights at once. Use $\Delta x = \frac{b - a}{n}$ as it stands. Then a curve below the axis gives a negative sum, and so does a reversed interval with a > b (lesson 2.5.3).

See the tests

import numpy as np


def test_left_four_strips():
    """Left sum of x² on [0, 1] with 4 strips is 0.21875"""
    expect_close(riemann(np.square, 0, 1, 4, "left"), 0.21875, what='riemann(x², 0, 1, 4, "left")')


def test_right_four_strips():
    """Right sum of x² on [0, 1] with 4 strips is 0.46875"""
    expect_close(riemann(np.square, 0, 1, 4, "right"), 0.46875, what='riemann(x², 0, 1, 4, "right")')


def test_midpoint_converges():
    """The midpoint sum of x² on [0, 1] with 1000 strips is within 1e-6 of 1/3"""
    expect_close(riemann(np.square, 0, 1, 1000), 1 / 3, what="midpoint sum, 1000 strips", abs_tol=1e-6, rel=0)


def test_other_interval():
    """Midpoint sum of eˣ on [0, 2] with 2000 strips is close to e² − 1"""
    expect_close(riemann(np.exp, 0, 2, 2000), np.e**2 - 1, what="midpoint sum of eˣ on [0, 2]", rel=1e-6)


def test_bad_rule():
    """An unknown rule raises ValueError"""
    try:
        riemann(np.square, 0, 1, 10, "top")
    except ValueError:
        return
    raise AssertionError('expected ValueError for rule="top"')


def test_below_axis():
    """A curve below the axis gives a negative sum: x − 1 on [0, 1] gives −0.5"""
    expect_close(riemann(lambda x: x - 1, 0, 1, 10), -0.5, what="midpoint sum of x − 1 on [0, 1]")


def test_reversed_interval():
    """Running from 1 back to 0 flips the sign: x² gives about −1/3"""
    expect_close(riemann(np.square, 1, 0, 1000), -1 / 3, what="riemann(x², 1, 0, 1000)", abs_tol=1e-6, rel=0)

Riemann sums

Antiderivatives

The fundamental theorem

Coding exercise

Free further reading