Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 4 − 1 2 x, y = 0, x = 1, x = 2; about the x-axis V = Sketch the region. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot Sketch the solid, and a typical disk or washer. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot

Answer 1

Answer:

${\dfrac{127 \pi}{12}$

Step-by-step explanation:

The formula for the volume for a solid of revolution about the x-axis on an interval [a,b] is

$\displaystyle V = \int_{a}^{b} \pi y^{2}dx$

If y =  4 - ½x, a = 1, and b = 2,

$\displaystyle V = \int_{1}^{2} \pi (4 - \frac{1}{2}x)^{2}dx = \pi \int_{1}^{2} \left(\dfrac{8-x }{2}\right)^{2}dx=\dfrac{\pi }{4}\int_{1}^{2} \left(8-x\right)^{2}dx\\\\=-\dfrac{\pi }{4}\times \dfrac{1}{3}\left[ 8 - x)^{3}\right]_{1}^{2}= -\dfrac{\pi }{12 }\left [512 - 192x + 24x^{2}-x^{3} \right ]_{1}^{2}\\\\=-\dfrac{\pi }{12}(512 - 384 + 96-8) + \dfrac{\pi }{12}(512 - 192 +24 -1)\\\\= -\dfrac{216\pi }{12} + \dfrac{343\pi }{12} = \mathbf{\dfrac{127 \pi}{12}\approx 33.25}$

The solid looks like the graph below.