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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.

Curve Hunter

Age 14 to 18
Challenge Level

Don't be afraid to experiment and sketch lots of rough curves before neatly drawing any new examples which you construct.

A continuous curve can be a straight line, or a collection of straight line segments which join together.

Some curves get closer and closer to the $x$-axis for large values of $x$, whereas othes get larger and larger as $x$ grows. 
You could classify your curves according to the number of zeros and/or number of points with zero gradient, along with other conditions that you might impose.