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QLDMath MethodsQuick questions
Unit 3: Further calculus and statistics
Quick questions on Applications of integration: area, average value and kinematics (QCE Mathematical Methods Unit 3)
13short 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 area under a single curve?Show answer
For $f(x) \geq 0$ on $[a, b]$:
What is area between two curves?Show answer
For two curves $y = f(x)$ and $y = g(x)$ on $[a, b]$ where $f(x) \geq g(x)$:
What is average value of a function?Show answer
The average value of $f$ on $[a, b]$ is
What is kinematics?Show answer
In rectilinear (straight-line) motion, displacement $s(t)$, velocity $v(t)$ and acceleration $a(t)$ are linked by differentiation and integration.
What is area under a curve?Show answer
Find the area under $y = e^x$ from $x = 0$ to $x = 1$.
What is area between curves?Show answer
Find the area enclosed by $y = x$ and $y = x^3$ in the first quadrant.
What is average value?Show answer
Find the average value of $f(x) = \sin x$ on $[0, \pi]$.
What is top minus bottom inverted?Show answer
If you integrate bottom minus top you get a negative result. Take absolute values or swap the order.
What is forgetting to split at intersections?Show answer
Curves that cross on the interval require multiple integrals with the order flipped on each piece.
What is mixing displacement and distance?Show answer
Displacement is the signed integral of velocity. Total distance is the integral of $|v(t)|$. PSMT and EA markers test this distinction frequently.
What is skipping initial conditions in kinematics?Show answer
Both integration constants ($C_1$ and $C_2$) must be evaluated from initial conditions, not left in the answer.
What is wrong factor on the average value?Show answer
The denominator is $b - a$, not $b$ alone. A common slip on a $[0, b]$ interval gives the right answer by coincidence and then the wrong answer everywhere else.
What is using the definite integral for area without checking sign?Show answer
$\int_a^b f \, dx$ is signed area. For geometric area you may need to split at the zeros of $f$ and absolute-value each piece.