MathDB

628

Part of 2010 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 628

Source:

7/8/2010
(1) Evaluate the following definite integrals.
(a) ∫0π2cos⁡2xsin⁡x dx\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx
(b) ∫0π2(π−2x)cos⁡x dx\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx
(c) ∫0π2xcos⁡3x dx\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx
(2) Let aa be a positive constant. Find the area of the cross section cut by the plane z=sin⁡θ (0≤θ≤π2)z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right) of the solid such that
x2+y2+z2≤a2,  x2+y2≤ax,  z≥0x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0
, then find the volume of the solid.
1984 Yamanashi Medical University entrance exam
Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them.
Thanks in advance.
kunny
calculusintegrationtrigonometrygeometryfunctioncalculus computations