Math  /  Calculus

QuestionThree regions are defined in the figure.
Find the volume generated by rotating the given region about the specified line. R1 about OA\mathscr{R}_{1} \text { about } O A \square

Studdy Solution
Evaluate the integral:
V=π019x2dx V = \pi \int_{0}^{1} 9x^2 \, dx
V=9π01x2dx V = 9\pi \int_{0}^{1} x^2 \, dx
V=9π[x33]01 V = 9\pi \left[ \frac{x^3}{3} \right]_{0}^{1}
V=9π(133033) V = 9\pi \left( \frac{1^3}{3} - \frac{0^3}{3} \right)
V=9π(13) V = 9\pi \left( \frac{1}{3} \right)
V=3π V = 3\pi
The volume of the solid generated by rotating the region R1 \mathscr{R}_1 about the line OA OA is:
3π \boxed{3\pi}

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