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Tough Integration Qn on Volume, R(x)= (10-x)(x^2)/10?

The profile of an experimental propeller is given by R(x)= (10-x)(x^2)/10 for 0

1) What volume, V(t), of water is swept out by the leading portion of the blade in one revolution? (Enter an expression in terms of t)

Note from the form of R(x) that R(x) is positive on (0, 10) and zero at 0, 10. So the volume integral is fairly simple.

1. V(t) = π int(0 to t) (R(x)^2) dx
= π int(0 to t) [(100 - 20x + x^2) (x^4) / 100] dx
= π [x^5 / 5 - (1/5) x^6 / 6 + (1/100) x^7 / 7] [0 to t]
= π (t^5 / 5 - t^6 / 30 + t^7 / 700)

2. This is just V(10), i.e.
π (10^5 / 5 - 10^6 / 30 + 10^7 / 700)
= 952.4 cm^3 to 4 s.f.

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