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NSWMaths Extension 1Quick questions

Calculus (ME-C1, C2, C3)

Quick questions on Volumes of revolution: discs about the x-axis and y-axis

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 the disc method (rotation about the x-axis)?
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Consider a region bounded by y=f(x)y = f(x), the x-axis, and the vertical lines x=ax = a and x=bx = b (with f(x)≥0f(x) \ge 0 on [a,b][a, b]).
What is rotation about the y-axis?
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For a region with x=g(y)x = g(y) on [c,d][c, d] (with g(y)≥0g(y) \ge 0), rotating about the y-axis gives discs of radius g(y)g(y) and thickness dydy. Volume:
What is the washer method (rotation about an axis, with inner curve)?
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If the region is bounded above by y=f(x)y = f(x) and below by y=h(x)y = h(x) (with 0≤h(x)≤f(x)0 \le h(x) \le f(x)) and rotated about the x-axis:
What is setup recipe?
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1. Sketch the region (this is essential, do not skip). 2. Choose the axis of rotation.
What is common pitfalls in setup?
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:::worked Worked example ### Simple disc, x-axis
What is disc, y-axis?
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Find the volume when y=x2y = x^2 between y=0y = 0 and y=4y = 4 is rotated about the y-axis.
What is washer?
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Find the volume when the region between y=x2y = x^2 and y=2xy = 2 x is rotated about the x-axis.
What is trig integrand?
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Find the volume when y=sin⁡xy = \sin x between x=0x = 0 and x=πx = \pi is rotated about the x-axis.
What is inverse-trig integrand?
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Find the volume when y=1x2+1y = \frac{1}{\sqrt{x^2 + 1}} between x=0x = 0 and x=1x = 1 is rotated about the x-axis.
What is wrong axis for the strip?
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Rotation about the x-axis uses vertical strips (dxdx); rotation about the y-axis uses horizontal strips (dydy).
What is forgetting to invert the function?
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Rotating y=f(x)y = f(x) about the y-axis requires xx as a function of yy, not the other way around.
What is limits in the wrong variable?
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If you switch from dxdx to dydy, the limits change to the y-range of the region.
What is sign error in the washer formula?
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Outer minus inner, both squared. Reversing the order or forgetting to subtract gives nonsense. :::

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