Question

use the method of disks to determine the volume of the solid by rotating the region bounded by y=10-2x,y=x+1 and y=7 about the

a. line x=8 b. line x=1 c. line x=-4

Answer 1

$$y=10-2x=>x=-0.5y+5$$

$$y=x+1=>x=y-1$$

$$y-1=-0.5y+5$$

$$y=4$$

a.

$$V=\displaystyle\int_{4}^{7}\pi(8+0.5y-5)^2dy$$ $$-\displaystyle\int_{4}^{7}\pi(8-y+1)^2dy$$

$$=\pi[-0.25y^3+10.5y^2-72y]\begin{matrix} 7\\ 4 \end{matrix}=$$

$$=\dfrac{243\pi}{4}({units}^3)$$

b.

$$V=\displaystyle\int_{4}^{7}\pi(y-1-1)^2dy$$ $$-\displaystyle\int_{4}^{7}\pi(-0.5y+54)^2dy$$

$$=\pi[0.25y^3-12y]\begin{matrix} 7\\ 4 \end{matrix}=$$

$$=\dfrac{135\pi}{4}({units}^3)$$

c.

$$V=\displaystyle\int_{4}^{7}\pi(y-1+4)^2dy$$ $$-\displaystyle\int_{4}^{7}\pi(-0.5y+5-1)^2dy$$

$$=\pi[0.25y^3+7.5y^2-72y]\begin{matrix} 7\\ 4 \end{matrix}=$$

$$=\dfrac{405\pi}{4}({units}^3)$$