use washer method to find volume of solid when region is bounded by y=sqrt(x), y

Question

use washer method to find volume of solid when region is bounded by y=sqrt(x), y=0 and x=9 is revolved about the y-axis

Explanation / Answer

y =vx intersects x-axis (line y=0) at point (0, 0) We integrate from x = 0 (intersection) to x = 1 (given) Axis of rotation: line = -4 Surface area of cylindrical shell = 2p r h where r = distance from axis of rotation = x - (-4) = x + 4 and h = height of cylinder = y = vx V = 2p ?0¹ r h dx V = 2p ?0¹ (x + 4) vx dx V = 2p ?0¹ (x^(3/2) + 32x^(1/2)) dx Solving, we get V = 92p/15 ========================= 3. Points of intersection: (0,0) and (4,4) y² = 4x ----> y = 2vx Cylinder has radius = x, height = 2vx - x V = 2p ?04 x (2vx - x) dx V = 2p ?04 (2x^(3/2) - x²) dx Solving, we get: V = 128p/15 ========================= 4. Points of intersection (0,0) and (1,5) Since we are rotating about a horizontal axis, we will have to integrate with respect to y We integrate from y = 0 to y = 5 y = 5x² ------> x = v(y/5) y = 5x -------> x = y/5 Cylinder has radius = y, height = v(y/5) - y/5 V = 2p ?05 y (v(y/5) - y/5) dy V = 2p ?05 (1/v5 y^(3/2) - 1/5 y²) dy V = 10p/3 -------------------- Of course, this would have been easier to solve using washer method Integrate from x = 0 to x = 1 Outer radius: R = 5x Inner radius: r = 5x² V = p ?0¹ (R² - r²) dx V = p ?0¹ (25x² - 25x4) dx V = 10p/3

Related Questions

Navigate

• Browse (All)
• Browse U
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.