QuestionFind the volume of the solid obtained by rotating the region enclosed by the graphs of $y=12-x, y=3 x-4$ and $x=0$ about the $y$-axis. $V=$ $\square$ Dreview Mv Answers Submit Answers

Studdy Solution
Evaluate the integral:
$$V = 2\pi \left[ \int_{0}^{4} 16x \, dx - \int_{0}^{4} 4x^2 \, dx \right]$$
Calculate each integral separately:
$$\int 16x \, dx = 8x^2 + C$$ $$\int 4x^2 \, dx = \frac{4}{3}x^3 + C$$
Evaluate from 0 to 4:
$$V = 2\pi \left[ (8(4)^2 - \frac{4}{3}(4)^3) - (8(0)^2 - \frac{4}{3}(0)^3) \right]$$
$$V = 2\pi \left[ (8 \cdot 16 - \frac{4}{3} \cdot 64) \right]$$
$$V = 2\pi \left[ 128 - \frac{256}{3} \right]$$
$$V = 2\pi \left[ \frac{384}{3} - \frac{256}{3} \right]$$
$$V = 2\pi \left[ \frac{128}{3} \right]$$
$$V = \frac{256\pi}{3}$$
The volume of the solid is:
$$V = \frac{256\pi}{3}$$