Calculus Question: volumes of solids. Answer this question using both the Washer Method and the Disk Method. The circle x^2 + (y-1)^2 = 1 is rotated about the line x=2. Find the volume.

Question

Answer this question using both the Washer Method and the Disk Method. The circle x^2 + (y-1)^2 = 1 is rotated about the line x=2. Find the volume.

Answer 1

By washer method: x = ±√(2y - y²); v = π∫₀²{[2 + √(2y - y²)]² - [2 - √(2y - y²)]²}dy = π∫₀²8√(2y - y²)dy= 39.478;

By shell method: y = 1 ± √(1 - x²); v = 2π∫_-1¹(2 - x)·2√1 - x²dx = 4π[∫_-1¹2√1 - x²dx - ∫_-1¹x√1- x²dx] = 4π² = 39.478

In 2nd line first integral equel π as area of circle radius 1, second integral is 0 as integral from odd function in symmetric limits. We get exact result v = 4π²