Applications of Integration : Exercise 2

Introduction
Exercise 2 on integration, will focus on, applying integration to find the volume of bounded region.

1. The finite region bounded by the curve with the equation y = 8x – x² and rotated through 360° about the x axis. Using integration find, in terms of π, find the volume of its solid form.


2. A solid is formed by rotating the part of the graph of y = 2x² between x = 1 and x = 2 through 360° about the x-axis. Find the volume of the solid. 


3. Find the volume generated by revolving the region bounded by y = x² + 2, y = 1, x = 0 and x = 2 about the x-axis.


4. Calculate, in terms of π, the volume generated by revolving the region through 360° about the y-axis.
   
   
5. Figure below shows the curve y = 9x -x² which intersects a straight line at point (3, 0) and point M. Calculate
   a) the coordinates of M
   b) the area of the shaded region
   c) the volume generated, in terms of π,
   when the region bounded by the curve, the y-axis and the straight line is revolved through 360° about the y-axis.