# Volume by rotating?

I've tried this so many times! Please Help

### 2 Answers

- PopeLv 71 month ago
I certainly would not use integration for this one. I often lament that students are held back in calculus because they do not first master the fundamentals of geometry.

Begin by rotating rectangle OABC about axis BC. That generates a right cylinder of radius 5 and length 1. Its volume is 25π.

Now remove the solid generated by regions R₂ and R₃. Together they generate the right cone having the same radius and length as the cylinder above. The cone has 1/3 the volume of the cylinder, so its removal leaves 2/3 of it.

(2/3)(25π) = 50π/3

- rotchmLv 71 month ago
Using disc method (washer) .

At x=x, draw a vertical rectangle in R1. Its height is thus 5x, and width dx.

Rotating it makes a washer of volume dV = π(5² - (5-5x)²) dx. Convince yourself of this; its just highschool geometry.

Now, just integrate form 0 to 1.

Done!