find the volume generated of the region bounded by the parabola y=x^2 and the line y=b when revolved (b>0):
a.) about the x-axis
b.) about the line y=-1
c.) about the line y=a, where a>b
To find the volume generated by the solid obtained from revolving the region bounded by the parabola y = x^2 and the line y = b, we can use the method of cylindrical shells.
a.) To find the volume when revolved about the x-axis:
- First, determine the points of intersection between the parabola y = x^2 and the line y = b. This can be done by setting x^2 = b and solving for x.
- The points of intersection are x = -sqrt(b) and x = sqrt(b).
- Now, consider a vertical strip of thickness Δx, located at some x between -sqrt(b) and sqrt(b).
- The height of the strip (h) is given by h = x^2 - b.
- The length of the strip (L) is given by L = 2πx (circumference of the shell).
- The volume of the shell (V) is given by V = L * h * Δx = 2πx(x^2 - b)Δx.
- To find the total volume, we integrate V with respect to x from -sqrt(b) to sqrt(b): V = ∫[sqrt(b),-sqrt(b)] 2πx(x^2 - b) dx.
b.) To find the volume when revolved about the line y = -1:
- First, shift the whole region 1 unit upwards by rewriting the parabola equation as y = x^2 + 1.
- Again, determine the points of intersection between the parabola y = x^2 + 1 and the line y = b + 1.
- The points of intersection are x = -sqrt(b + 1) and x = sqrt(b + 1).
- Similarly, consider a vertical strip of thickness Δx, located at some x between -sqrt(b + 1) and sqrt(b + 1).
- The height of the strip (h) is given by h = x^2 + 1 - (b + 1) = x^2 - b.
- The length of the strip (L) is L = 2πx.
- The volume of the shell (V) is given by V = L * h * Δx = 2πx(x^2 - b)Δx.
- To find the total volume, we integrate V with respect to x from -sqrt(b + 1) to sqrt(b + 1): V = ∫[sqrt(b + 1),-sqrt(b + 1)] 2πx(x^2 - b) dx.
c.) To find the volume when revolved about the line y = a (where a > b):
- Similarly, determine the points of intersection between the parabola y = x^2 and the line y = b.
- The points of intersection are x = -sqrt(b) and x = sqrt(b).
- Consider a vertical strip of thickness Δx, located at some x between -sqrt(b) and sqrt(b).
- The distance between the line y = a and the parabola y = x^2 is given by h = a - x^2.
- The length of the strip (L) is L = 2πx.
- The volume of the shell (V) is given by V = L * h * Δx = 2πx(a - x^2)Δx.
- To find the total volume, we integrate V with respect to x from -sqrt(b) to sqrt(b): V = ∫[sqrt(b),-sqrt(b)] 2πx(a - x^2) dx.
By following these steps, you can calculate the volume generated for each scenario.