Final answer:
To find the volume of the solid created by revolving a bounded region defined by two lines and a vertical axis around the y-axis, one must sketch the region, establish the bounds, use the disc method to set up the integral, and then perform the integration from the lower to the upper bounds of y.
Step-by-step explanation:
The problem involves sketching the region bounded by the curves x+y=1, 4x+y=4, and x=0, followed by calculating the volume of the solid generated when this region is revolved around the y-axis.
To begin, solve the equations of the curves to determine the points of intersection and draw the bounded region on the coordinate plane. With x set to zero, y equals 1 for the first line and y equals 4 for the second line, indicating that these are the limits for our region.
To find the volume of the revolution about the y-axis, we can use the disc method, calculating the volume of each infinitesimal disc that is formed by revolving the strip about the y-axis. The volume of each disc is πr²h, where r is the radius (the x-value of the bounding line) and h is the infinitesimal thickness (dy).
Integrating from y=1 to y=4, the volume V is given by the integral ∫π(x²)dy over the interval [1, 4], using the relation
x = 4-y from the first equation and x=1-(y/4) from the second equation to replace x in the integral.
Finally, integrate the expression for the volume to find the exact numerical value. The result will provide the volume of the solid formed by the revolution of the sketched region around the y-axis.