Integration · 积分
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| integration/ˌɪntɪˈɡreɪʃn/ | 积分 | jī fēn |
| constant of integration/ˈkɒnstənt ɒv ˌɪntɪˈɡreɪʃn/ | 积分常数 | jī fēn cháng shù |
| definite integral/ˈdefɪnət ˈɪntɪɡrəl/ | 定积分 | dìng jī fēn |
| volume of revolution/ˈvɒljuːm ɒv ˌrevəˈluːʃn/ | 旋转体体积 | xuán zhuǎn tǐ tǐ jī |
The area under the curve
- How much fuel does a rocket burn if its thrust varies over time? How much land does a curved field cover?
- Integration 积分 answers these questions — it finds the total accumulated quantity from a rate of change.
曲线下的面积
- 如果一枚火箭的推力随时间变化,它烧多少燃料?一块弯曲的田地覆盖多少土地?
- 积分(integration)回答这些问题——它从一个变化率找到总的累积量。
Indefinite integration
- Integration reverses differentiation:
- The $+\,C$ is the constant of integration 积分常数 — find it from a known point.
Worked example. $\int 3x^2\,dx = x^3 + C$. If the curve passes through $(1, 5)$: $1^3 + C = 5$, so $C = 4$ and $y = x^3 + 4$.
A definite integral 定积分 measures the shaded area between the curve and the x-axis
不定积分
- 积分逆转微分:
- 那个 $+\,C$ 是积分常数(constant of integration)——从一个已知的点找到它。
算例。 $\int 3x^2\,dx = x^3 + C$。如果曲线过 $(1, 5)$:$1^3 + C = 5$,所以 $C = 4$ 而 $y = x^3 + 4$。
一个定积分测量曲线和 x 轴之间的阴影面积
The area under the curve · 曲线下的面积
area = ∫ f(x) dx
Drag the limits a and b — the shaded area · 面积 between the curve and the x-axis is the definite integral. · 拖动上下限 a 和 b——曲线和 x 轴之间的阴影面积就是定积分。
Why does an indefinite integral include "+ C"? · 为什么一个不定积分包括 "+ C"?
Differentiating a constant gives 0, so the original could have had any constant — hence + C. · 微分一个常数给出 0,所以原来的可能有任何常数——因此 + C。
If dy/dx = 4x and the curve passes through (1, 7), what is C in y = 2x² + C? · 如果 dy/dx = 4x 而曲线过 (1, 7),y = 2x² + C 中的 C 是多少?
y = 2x² + C. At (1, 7): 7 = 2(1)² + C → C = 5. · y = 2x² + C。在 (1, 7):7 = 2(1)² + C → C = 5。
Definite integrals
- A definite integral gives a number: $\displaystyle\int_p^q f(x)\,dx = F(q) - F(p)$.
- The area under a curve from $p$ to $q$ is $\displaystyle\int_p^q y\,dx$ (for two curves, integrate top − bottom).
The definite integral $\int_0^3 x^2\,dx = \left[\dfrac{x^3}{3}\right]_0^3 = 9 - 0 = 9$. The shaded area is $9$ square units.
Area below the x-axis is negative. If the curve dips below the axis, the integral gives a negative value. To find the total area, split at the roots and take the absolute value of each part.
定积分
- 一个定积分(definite integral)给出一个数:$\displaystyle\int_p^q f(x)\,dx = F(q) - F(p)$。
- 从 $p$ 到 $q$ 的曲线下的面积是 $\displaystyle\int_p^q y\,dx$(对两条曲线,积分 上 − 下)。
定积分 $\int_0^3 x^2\,dx = \left[\dfrac{x^3}{3}\right]_0^3 = 9 - 0 = 9$。阴影面积是 $9$ 平方单位。
x 轴下面的面积是负的。 如果曲线降到轴下面,积分给出一个负值。要找到总面积,在根处分开并取每一部分的绝对值。
Evaluate the definite integral of 2x from 0 to 3 (∫₀³ 2x dx). · 求 2x 从 0 到 3 的定积分(∫₀³ 2x dx)。
∫2x dx = x², so [x²]₀³ = 9 − 0 = 9. · ∫2x dx = x²,所以 [x²]₀³ = 9 − 0 = 9。
Evaluate ∫₀² 3x² dx. · 求 ∫₀² 3x² dx。
∫3x² dx = x³, so [x³]₀² = 8 − 0 = 8. · ∫3x² dx = x³,所以 [x³]₀² = 8 − 0 = 8。
If a curve is below the x-axis, the definite integral gives a negative value. · 如果一条曲线在 x 轴下面,定积分给出一个负值。
Area below the x-axis contributes negatively to the integral. Take the absolute value to find the actual area. · x 轴下面的面积对积分贡献为负。取绝对值来求实际的面积。
Volume of revolution 旋转体体积
- Volume of revolution about the $x$-axis: $V = \pi\displaystyle\int_p^q y^2\,dx$.
- Think of it as stacking infinitely thin discs of radius $y$ and thickness $dx$.
Rotating the region under a curve around the x-axis sweeps out a solid, summed as thin discs
旋转体的体积
- 关于 $x$ 轴的旋转体的体积(volume of revolution):$V = \pi\displaystyle\int_p^q y^2\,dx$。
- 把它想成堆叠半径 $y$、厚度 $dx$ 的无穷薄的圆盘。
把曲线下的区域绕 x 轴旋转扫出一个立体,作为薄圆盘求和
The volume of revolution of y = x about the x-axis from 0 to 2 is π∫₀² x² dx. Find the value (without π). · y = x 关于 x 轴从 0 到 2 的旋转体的体积是 π∫₀² x² dx。求这个值(不含 π)。
∫₀² x² dx = [x³/3]₀² = 8/3 ≈ 2.67. · ∫₀² x² dx = [x³/3]₀² = 8/3 ≈ 2.67。
Worked example
- Find the area under $y = x^2$ from $x = 0$ to $x = 3$:
- $\displaystyle\int_0^3 x^2\,dx = \left[\dfrac{x^3}{3}\right]_0^3 = \dfrac{27}{3} - 0 = 9$.
- An improper integral has an infinite limit or an unbounded integrand.
算例
- 找 $y = x^2$ 从 $x = 0$ 到 $x = 3$ 的曲线下面积:
- $\displaystyle\int_0^3 x^2\,dx = \left[\dfrac{x^3}{3}\right]_0^3 = \dfrac{27}{3} - 0 = 9$。
- 反常(improper)积分的上下限为无穷,或被积函数无界。
You've got it
- integration reverses differentiation; remember the $+\,C$
- a definite integral $\int_p^q f(x)\,dx = F(q) - F(p)$ = the area under the curve
- volume of revolution about the $x$-axis $= \pi\int y^2\,dx$
- area below the axis is negative — split and take absolute values for total area
你掌握了
- 积分逆转微分;记住 $+\,C$
- 一个定积分 $\int_p^q f(x)\,dx = F(q) - F(p)$ = 曲线下的面积
- 关于 $x$ 轴的旋转体的体积 $= \pi\int y^2\,dx$
- 轴下面的面积是负的——分开并取绝对值来求总面积