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VICMath MethodsQuick questions
Unit 4
Quick questions on Definite integration and the fundamental theorem: VCE Math Methods Unit 4
11short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is example 1. Polynomial?Show answer
$\int_{0}^{2} (x^3 - 3 x) \, dx$.
What is example 2. Exponential?Show answer
Antiderivative. $F(x) = \frac{1}{2} e^{2x}$.
What is example 3. Logarithm?Show answer
$\int_{1}^{2} \frac{1}{x} \, dx = [\ln\lvert x \rvert]_{1}^{2} = \ln 2 - \ln 1 = \ln 2$.
What is example 4. Symmetric circular?Show answer
$\int_{-\pi}^{\pi} \sin(x) \, dx$.
What is example 5. Interval splitting?Show answer
Given $\int_{0}^{5} f(x) \, dx = 12$ and $\int_{3}^{5} f(x) \, dx = 4$, find $\int_{0}^{3} f(x) \, dx$.
What is forgetting to subtract?Show answer
Writing $F(b) - F(a)$ as $F(b)$ alone or as $F(a) - F(b)$ (wrong sign). Always evaluate at upper minus lower.
What is wrong antiderivative?Show answer
Especially the $\frac{1}{k}$ factor for $e^{kx}$, $\sin(kx)$, $\cos(kx)$. Forgetting it makes the answer off by a factor.
What is bracket discipline?Show answer
$F(b) - F(a)$ where $F$ has multiple terms requires brackets around the substituted expressions. Without brackets, sign errors cascade.
What is including the constant of integration?Show answer
For definite integrals, $c$ cancels and is not written. Including it earns no marks and signals a procedural slip.
What is treating signed area as area?Show answer
$\int_{a}^{b} f(x) \, dx$ is signed area. For total area (always non-negative), see the area-under-curves dot point. Confusing the two is a common error.
What is substituting $\pi$ as 3.14 in Paper 1?Show answer
Paper 1 expects exact values. Use $\sin(\pi/2) = 1$, $\cos(\pi) = -1$ exactly.