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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?

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

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.

Scientific Curves

Age 16 to 18
Challenge Level

This problem is aimed at interested and enthusiastic students: although we don't recommend it for mainstream classroom use it provides enrichment and will develop mathematical thinking and knowledge.

Like all NRICH problems it has been carefully designed to be full of mathematical interest and as such might be used in a variety of ways: you might suggest that keen students consider the problem in their own time, use it for a maths club at school or use it in class as a vehicle to foster discussion and mathematical thinking. See our Teachers Guide to Getting Started with rich tasks for more suggestions and ideas.
This problem will be of particular interest to students considering a university course containing an element of mathematical physics.