John works for TabbySoft selling computer software. He earns 3% on the first $10,000 sold
and 4% on all sales over $10,000. Last week
John's sales totaled $34,216.46. Determine his
total commission for last week.

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Ответ:

AlexGaming1822

27.03.2020

Mathematics

I assume you're revolving the region with those bounds about the x-axis, and supposed to find the volume.

Via the disk method,

$\displaystyle\pi\int_0^\pi(3\sin2x)^2\,\mathrm dx=9\pi\int_0^\pi\sin^22x\,\mathrm dx$

Recall the half-angle identity for sine:

$\sin^2t=\dfrac{1-\cos2t}2$
$\implies\displaystyle\frac{9\pi}2\int_0^\pi(1-\cos4x)\,\mathrm dx$
$=\displaystyle\frac{9\pi}2\left(x-\frac14\sin4x\right)\bigg|_{x=0}^{x=\pi}$
$=\dfrac{9\pi^2}2$

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John works for TabbySoft selling computer software. He earns 3% on the first $10,000 sold and 4% on all sales over $10,000. Last week John's sales totaled $34,216.46. Determine his total commission for last week.