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NSWMaths AdvancedQuick questions
Year 12: Calculus
Quick questions on Areas between curves and volumes of revolution using definite integrals
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 area under a curve?Show answer
If $f(x) \geq 0$ on $[a, b]$, then the area between $y = f(x)$ and the $x$-axis is
What is area between two curves?Show answer
If $f(x) \geq g(x)$ on $[a, b]$, the area between them is
What is area between a curve and the $y$-axis?Show answer
If the boundary is described as $x = h(y)$, integrate with respect to $y$:
What is volume of revolution about the $x$-axis?Show answer
Rotating the region under $y = f(x)$ between $x = a$ and $x = b$ about the $x$-axis produces a solid whose cross sections perpendicular to the axis are disks of radius $f(x)$. The volume is
What is volume of revolution about the $y$-axis?Show answer
Rotating the region between $x = h(y)$ and the $y$-axis, between $y = c$ and $y = d$, about the $y$-axis gives
What is volume between two curves?Show answer
If a region is bounded by $y = f(x)$ above and $y = g(x)$ below, rotating about the $x$-axis gives a washer with outer radius $f(x)$ and inner radius $g(x)$:
What is forgetting the absolute value?Show answer
A definite integral can be negative if the curve dips below the axis. Area is non-negative, so split or take absolute values.
What is squaring the difference instead of differencing the squares?Show answer
Volume of a washer uses $[f(x)]^2 - [g(x)]^2$, not $(f(x) - g(x))^2$.
What is integrating with respect to the wrong variable?Show answer
Rotation about the $x$-axis pairs with $\int f(x)^2 \, dx$. Rotation about the $y$-axis pairs with $\int h(y)^2 \, dy$.
What is missing intersections?Show answer
If the curves cross inside your interval, the upper and lower roles swap. Split the integral and re-test which curve is on top in each subinterval.
What is dropping $\pi$?Show answer
A volume of revolution always carries a factor of $\pi$. Forgetting it costs marks.