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QLDSpecialist MathematicsQuick questions

Unit 4: Further calculus, and statistical inference

Quick questions on Areas between curves and volumes of revolution in QCE Specialist Mathematics Unit 4

6short 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 are area between two curves?
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If f(x)≥g(x)f(x) \geq g(x) on [a,b][a, b], the area enclosed between them is
What is volume of revolution about the x-axis?
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Rotating the region under y=f(x)y = f(x) from x=ax = a to x=bx = b about the xx-axis sweeps out a solid of circular cross-sections of radius f(x)f(x). Each thin disc has volume π[f(x)]2 dx\pi [f(x)]^2\, dx, so
What is volume of revolution about the y-axis?
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Rotating the region between x=g(y)x = g(y) and the yy-axis from y=cy = c to y=dy = d about the yy-axis gives
What is region between two curves rotated about an axis?
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When the rotated region lies between an outer curve R(x)R(x) and an inner curve r(x)r(x), the cross-section is an annulus (washer):
What is identify the radius?
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The boundary curve is y=xy = \sqrt{x}, so the disc radius at position xx is f(x)=xf(x) = \sqrt{x}, and the region runs from x=0x = 0 to x=4x = 4.
What is check the scale?
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The solid is a paraboloid bounded by a disc of radius 22 at x=4x = 4. Its volume 8π≈25.18\pi \approx 25.1 is less than the enclosing cylinder π(2)2(4)=16π\pi (2)^2 (4) = 16\pi, as expected for a solid tapering to a point at the origin.

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