Other Applications of Integration

Question

1. The area bounded by the parabola ay = x² and the line y = b is revolve about the x − axis. Find the volume generated. (use cylindrical shell method)

2. How far from the x-axis and y-axis is the centroid of the area bounded by the curve y² = 8x, the y-axis and the line y = 4.

3. By iterated integration, find the area bounded by the parabola y² =4ax and its latus rectum.

Answer 1

V = ∫^y=b_y=0 2πy2√y dy = 4π ∫^y=b_y=0 y^3/2 dy = 4π (2/5) y^5/2 |^b₀ =(8π/5) b²√b.

y = ( ∫⁴₀yƒ(y)dy ) / ( ∫⁴₀ƒ(y) dy ) = ( ∫⁴₀y (y²/8)dy ) / ( ∫⁴₀(y²/8)dy = 3

x = ( ∫⁴₀(1/2) (ƒ(y))²dy⁾ / ( ∫⁴₀ƒ(y) dy ) = 3/5

A= ∫^2a_-2a ( a- y²/4a ) dy =[ ay - y³/12a ] |^2a_-2a = 8a²/3