Find the volume of revolution

Question

Calculate the volume of the solid generated by rotating the region $\{x=1,y=2,y=x\}$ along the y-axis.

My problem:My region is a triangle,how I can calculate this?I know the formula for curvilinear regions not triangle.Please help me.

Answer 1

We can use shell method :

$$ V=\int_a^b 2\pi x f(x)\ dx $$

This revolution-solid can be cut into shells : $$ Height=f(x),\ radius=x,\ Volume\ of\ a\ shell = 2\pi x f(x)\ \Delta x $$

Here $$ a=1 \leq x\leq b=2,\ f(x)= 2-x $$ so that $$ V=\int_1^2 2\pi x (2-x)\ dx =\frac{4\pi}{3}$$

Answer 2

Hint: What would be the volume of the region bounded by $x=0,x=1,y=1,y=2$ rotated about the $y$-axis? What about the volume of the region bounded by $x=0,y=x,y=1,y=2$? How can you use this to find your answer?