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Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

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Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

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Fractional Calculus I

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

Turning to Calculus

Age 16 to 18 Challenge Level:

The gradients and signs of functions are strongly related for any nice, smooth curve which you can draw. To get into the problem you can sketch small line segments and ask yourself: how can these be smoothly joined together?
Complications can arise for curves which turn suddenly or have gaps or become infinite at various points. Maybe you could think about creative ways to get from one place to the next.