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# Integration (P3 Revision) watch

1. Hi,

just unsure on tackling this qu:

a
∫√(a^2 - x^2) dx = (a^2)/2 . (Pi/3 - √3/4)
0.5a

Prove that this is true using the substitution x = asinθ

many thanks

streety
2. Use the substitution x=asinθ
3. It comes down to integrating cos^2(theta), which you can do by using cos(2theta) = 2cos^2(theta) - 1.
4. (Original post by streetyfatb)
Hi,

just unsure on tackling this qu:

a
∫√(a^2 - x^2) dx = (a^2)/2 . (Pi/3 - √3/4)
0.5a

Prove that this is true using the substitution x = asinθ

many thanks

streety
∫√(a^2 - x^2) dx = a∫√(1 - (x/a)^2) dx

substitute: x = asinθ
=>dx/dθ = acosθ

Hence, integral becomes:

a^2∫(Costheta)^2 dθ

Using a well known identity, we get:

a^2∫ (Cos2θ + 1)/2 dθ

=> a^2[ (Sin2theta)/4 + θ/2 ] (between a, and a/2)
=> a^2[ (x√(1 - (x/a)^2))/2a + arcsin(x/a)/2 ] (between a, and a/2)
=> a^2[ Pi/4 - √3/8 - Pi/12 ]
=> a^2[ Pi/6 - √3/8 ]
=> (a^2)/2[ Pi/3 - √3/4 ]

which is the required result.

Euclid

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