第5章積分

Key Concepts重要な概念

The definite integral can be used to calculate net signed area, which is the area above the x-axis less the area below the x-axis. Net signed area can be positive, negative, or zero.
定積分で符号付き面積を計算できる。これは x 軸より上の面積から下の面積を引いた値であり、正・負・0のいずれにもなる。
The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
定積分を構成する要素は、被積分関数、積分変数、積分区間の端点である。
Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
閉区間上の連続関数は積分可能である。連続でない関数も、不連続の性質によっては積分可能である。
The properties of definite integrals can be used to evaluate integrals.
定積分の性質を使って、積分の値を求められる。
The area under the curve of many functions can be calculated using geometric formulas.
多くの関数について、曲線の下の面積は幾何学の公式から求められる。
The average value of a function can be calculated using definite integrals.
定積分を使って、関数の平均値を計算できる。

Key Equations重要な式

Definite Integral

定積分

\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x

Properties of the Definite Integral

定積分の性質

\int_{a}^{a} f (x) d x = 0

\int_{b}^{a} f (x) d x = − \int_{a}^{b} f (x) d x

\int_{a}^{b} [f (x) + g (x)] d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x

\int_{a}^{b} [f (x) - g (x)] d x = \int_{a}^{b} f (x) d x - \int_{a}^{b} g (x) d x

\int_{a}^{b} c f (x) d x = c \int_{a}^{b} f (x)

for constant c

\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x

\int_{a}^{a} f (x) d x = 0

\int_{b}^{a} f (x) d x = − \int_{a}^{b} f (x) d x

\int_{a}^{b} [f (x) + g (x)] d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x

\int_{a}^{b} [f (x) - g (x)] d x = \int_{a}^{b} f (x) d x - \int_{a}^{b} g (x) d x

\int_{a}^{b} c f (x) d x = c \int_{a}^{b} f (x)

（c は定数）

\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x

Exercises 60–63練習問題 60–63

In the following exercises, express the limits as integrals.

次の各問では、極限を積分として表せ。

Exercise 60問題 60

lim_{n \to \infty} \sum_{i = 1}^{n} (x_{i}^{*}) Δ x

over

[1, 3]

区間

[1, 3]

における

lim_{n \to \infty} \sum_{i = 1}^{n} (x_{i}^{*}) Δ x

。

Exercise 61問題 61

lim_{n \to \infty} \sum_{i = 1}^{n} (5 {(x_{i}^{*})}^{2} - 3 {(x_{i}^{*})}^{3}) Δ x

over

[0, 2]

区間

[0, 2]

における

lim_{n \to \infty} \sum_{i = 1}^{n} (5 {(x_{i}^{*})}^{2} - 3 {(x_{i}^{*})}^{3}) Δ x

。

Solution解き方

\int_{0}^{2} (5 x^{2} - 3 x^{3}) d x

\int_{0}^{2} (5 x^{2} - 3 x^{3}) d x

Exercise 62問題 62

lim_{n \to \infty} \sum_{i = 1}^{n} {sin}^{2} (2 π x_{i}^{*}) Δ x

over

[0, 1]

区間

[0, 1]

における

lim_{n \to \infty} \sum_{i = 1}^{n} {sin}^{2} (2 π x_{i}^{*}) Δ x

。

Exercise 63問題 63

lim_{n \to \infty} \sum_{i = 1}^{n} {cos}^{2} (2 π x_{i}^{*}) Δ x

over

[0, 1]

区間

[0, 1]

における

lim_{n \to \infty} \sum_{i = 1}^{n} {cos}^{2} (2 π x_{i}^{*}) Δ x

。

Solution解き方

\int_{0}^{1} {cos}^{2} (2 π x) d x

\int_{0}^{1} {cos}^{2} (2 π x) d x

Exercises 64–69練習問題 64–69

In the following exercises, given L_n or R_n as indicated, express their limits as

n \to \infty

as definite integrals, identifying the correct intervals.

次の各問では、指定された L_n または R_n について、

n \to \infty

のときの極限を定積分で表せ。積分区間を正しく定めること。

Exercise 64問題 64

L_{n} = \frac{1}{n} \sum_{i = 1}^{n} \frac{i - 1}{n}

L_{n} = \frac{1}{n} \sum_{i = 1}^{n} \frac{i - 1}{n}

Exercise 65問題 65

R_{n} = \frac{1}{n} \sum_{i = 1}^{n} \frac{i}{n}

R_{n} = \frac{1}{n} \sum_{i = 1}^{n} \frac{i}{n}

Solution解き方

\int_{0}^{1} x d x

\int_{0}^{1} x d x

Exercise 66問題 66

L_{n} = \frac{2}{n} \sum_{i = 1}^{n} (1 + 2 \frac{i - 1}{n})

L_{n} = \frac{2}{n} \sum_{i = 1}^{n} (1 + 2 \frac{i - 1}{n})

Exercise 67問題 67

R_{n} = \frac{3}{n} \sum_{i = 1}^{n} (3 + 3 \frac{i}{n})

R_{n} = \frac{3}{n} \sum_{i = 1}^{n} (3 + 3 \frac{i}{n})

Solution解き方

\int_{3}^{6} x d x

\int_{3}^{6} x d x

Exercise 68問題 68

L_{n} = \frac{2 π}{n} \sum_{i = 1}^{n} 2 π \frac{i - 1}{n} cos (2 π \frac{i - 1}{n})

L_{n} = \frac{2 π}{n} \sum_{i = 1}^{n} 2 π \frac{i - 1}{n} cos (2 π \frac{i - 1}{n})

Exercise 69問題 69

R_{n} = \frac{1}{n} \sum_{i = 1}^{n} (1 + \frac{i}{n}) log ({(1 + \frac{i}{n})}^{2})

R_{n} = \frac{1}{n} \sum_{i = 1}^{n} (1 + \frac{i}{n}) log ({(1 + \frac{i}{n})}^{2})

Solution解き方

\int_{1}^{2} x log (x^{2}) d x

\int_{1}^{2} x log (x^{2}) d x

Exercises 70–75練習問題 70–75

In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis.

次の各問では、図示された関数の積分を、三角形と円の面積公式で求めよ。x 軸より下の面積は引くこと。

Exercise 70問題 70