# DBT JRF 2022 — Question 19

Exam: DBT JRF 2022 · Section: A · Topic: Aptitude › Calculus - Volume of Revolution · Difficulty: Medium

## Question

The line y = 2x from x = 0 to 1 is rotated around the y-axis. The volume generated is:

## Options

- **A.** $\frac{1}{2}\pi$
- **B.** $\frac{2}{3}\pi$
- **C.** $\frac{3}{4}\pi$
- **D.** $\pi$

## Answer

Option B

## Explanation

Volume of revolution — rotating y = 2x about the y-axis: When rotating the line $y = 2x$ (from x = 0 to 1) about the y-axis, express x in terms of y: $x = \frac{y}{2}$ , and y ranges from 0 to 2. Using the disk method (integrating with respect to y): $V = \pi \int_0^2 x^2\,dy = \pi \int_0^2 \left(\frac{y}{2}\right)^2 dy = \frac{\pi}{4} \int_0^2 y^2\,dy = \frac{\pi}{4} \left[\frac{y^3}{3}\right]_0^2 = \frac{\pi}{4} \cdot \frac{8}{3} = \frac{2\pi}{3}$ This generates a cone with base radius 1 and height 2. Answer: $\dfrac{2}{3}\pi$ (Option 2).

Source: https://mypyqbuddy.com/questions/the-line-y-2x-from-x-0-to-6a104cd5920734408bc0c6f0