How to find the volume if the shown area is rotated around the $y$-axis?
The area will be bounded by $=^2+1$, $y=2x$ and $x=0$.
2026-03-30 08:35:59.1774859759
Volume by Rotation Using Integration
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Hint: $V_y=\pi\int_c^d x^2 dy$.
So, $V_{y_1}=\pi\int_0^2 \left(\frac{y}{2}\right)^2dy$ is the volume of cone, and the $V_{y_2}=\pi\int_1^2\left( y-1 \right)dy$ is the volume of a rotationally symmetric body given by a rotation of parabola $x^2=y-1$ around $y$-axis. Hence, desired volume is:
$$V_{y_1} - V_{y_2}.$$