A 14 foot ladder is leaned up against a building. The ladder reaches a height of 2 squared 15 feet up the building. Find the dis
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Integrating with shells is the easier method.
<em>V</em> = 2<em>π</em> ∫₁³ <em>x</em> (√<em>x</em> + 3<em>x</em>) d<em>x</em>
That is, at various values of <em>x</em> in the interval [1, 3], we take <em>n</em> shells of radius <em>x</em>, height <em>y</em> = √<em>x</em> + 3<em>x</em>, and thickness ∆<em>x</em> so that each shell contributes a volume of 2<em>π</em> <em>x</em> (√<em>x</em> + 3<em>x</em>) ∆<em>x</em>. We then let <em>n</em> → ∞ so that ∆<em>x</em> → d<em>x</em> and sum all of the volumes by integrating.
To compute the integral, just expand the integrand:
<em>V</em> = 2<em>π</em> ∫₁³ (<em>x </em>³ʹ² + 3<em>x</em> ²) d<em>x</em>
<em>V</em> = 2<em>π</em> (2/5 <em>x </em>⁵ʹ² + <em>x</em> ³) |₁³
<em>V</em> = 2<em>π</em> ((2/5 ×<em> </em>3⁵ʹ² + 3³) - (2/5 × 1⁵ʹ² + 1³))
<em>V</em> = 4<em>π</em>/5 (9√3 + 64)
