▶️ Answer/Explanation
Solution
Correct Answer: E
Volume of revolution is calculated using the disk method: V = π∫[f(x)]²dx from a to b
Here, V = π∫(2 + sinx)²dx from 0 to 5
Expanding: V = π∫(4 + 4sinx + sin²x)dx from 0 to 5
Using a calculator to evaluate this integral gives approximately 80.115 cubic inches
This represents the volume of the vase formed by rotating the curve about the x-axis
▶️ Answer/Explanation
Solution
Correct Answer: E
To find the volume when revolved about the y-axis, we use the disk method: \( V = \pi \int_{a}^{b} [f(y)]^2 dy \)
Here, \( f(y) = \sqrt{y – 2} \), and the bounds are from y=2 to y=5 (since at x=0, y=2)
The integral becomes: \( V = \pi \int_{2}^{5} (y – 2) dy = \pi \left[ \frac{y^2}{2} – 2y \right]_{2}^{5} \)
Evaluating gives: \( \pi \left( \frac{25}{2} – 10 – (2 – 4) \right) = \pi \left( 2.5 + 2 \right) = 4.5\pi \approx 14.137 \)
Therefore, the correct answer is E.