C11 Integration
对应大纲的内容
| MM7.1 | Definite integration as related to the ‘area between a curve and an axis’. The difference between finding a definite integral and finding the area between a curve and an axis is expected to be understood. |
| MM7.2 | Finding definite and indefinite integrals of $x^{n}$ for $n$ rational, $n \neq -1$, and related sums and differences, including expressions which require simplification prior to integrating.
For example, $\int (x + 2)^{2}\,\mathrm{d}x$ and $\int \frac{(3x – 5)^{2}}{x^{\frac{1}{2}}}\,\mathrm{d}x$. |
| MM7.4 | Combining integrals with either equal or contiguous ranges.
For example, $\int_{2}^{5} f(x)\,\mathrm{d}x + \int_{2}^{5} g(x)\,\mathrm{d}x = \int_{2}^{5} [f(x) + g(x)]\,\mathrm{d}x$, and $\int_{4}^{2} f(x)\,\mathrm{d}x$ $+$ $\int_{4}^{3} f(x)\,\mathrm{d}x$ $=$ $\int_{2}^{3} f(x)\,\mathrm{d}x$. |
| MM7.5 | Approximation of the area under a curve using the trapezium rule; determination of whether this constitutes an overestimate or an underestimate. |
| MM7.6 | Solving differential equations of the form $\frac{\mathrm{d}y}{\mathrm{d}x} = f(x)$. |
学习目标
- Master pre-integration algebra by simplifying complex rational functions and nested polynomials (e.g., expanding $\frac{(3 x-5)^2}{x^{\frac{1}{2}}}$ into linear combinations of $x^n$ to apply the power rule directly, bypassing advanced integration techniques.
- Navigate the area vs. integral dichotomy by distinguishing between the algebraic evaluation of a definite integral and the total physical area bounded by a curve, splitting domains at -axis roots to ensure positive area results.
- Exploit contiguous boundary logic by manipulating integral limits using range-splitting $\left(\int_a^c \square\right.$ $+$ $\int_c^b \square$ $=$ $\left.\int_a^b \square\right)$ and boundary-inversion $\left(\int_a^b \square=-\int_b^a \square\right)$ to solve complex problems even when the function is unknown.Evaluate trapezium rule bounds by using curve concavity $\left(f^{\prime \prime}(x)\right)$ to determine if an approximation is an over- or underestimate, and predicting how functional transformations (e.g., $1-f(x)$ or $2 f(x)$ ) affect estimation error.
- Solve boundary-value differential equations by integrating $\frac{\mathrm{d} y}{\mathrm{~d} x}=f(x)$to establish general solutions, then isolating the constant of integration ( $C$ ) using provided boundary coordinates.
建议时间
| Task | No. of Questions | Suggested Time |
|---|---|---|
| Examples | 15 | 200 min |
| Quiz | 11 | 30 min |
| Practice A | 30 | 60 min |
| Practice B | 30 | 75 min |
| Total | 84 | approx. 6-7 hrs |
Practice:
Practice:
