Integration
| English | Chinese | Pinyin |
|---|---|---|
| integration | 积分 | jī fēn |
| constant of integration | 积分常数 | jī fēn cháng shù |
| definite integral | 定积分 | dìng jī fēn |
| volume of revolution | 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.
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
The area under the curve
area = ∫ f(x) dx
Drag the limits a and b — the shaded area between the curve and the x-axis is the definite integral.
Why does an indefinite integral include "+ C"?
Differentiating a constant gives 0, so the original could have had any constant — hence + C.
If dy/dx = 4x and the curve passes through (1, 7), what is C in y = 2x² + C?
y = 2x² + C. At (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.
Evaluate the definite integral of 2x from 0 to 3 (∫₀³ 2x dx).
∫2x dx = x², so [x²]₀³ = 9 − 0 = 9.
Evaluate ∫₀² 3x² dx.
∫3x² dx = x³, so [x³]₀² = 8 − 0 = 8.
If a curve is below the x-axis, the definite integral gives a negative value.
Area below the x-axis contributes negatively to the integral. Take the absolute value to find the actual area.
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
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).
∫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.
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