A manufacturer designs a basin whose cross-sectional profile (in centimeters) is…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A manufacturer designs a basin whose cross-sectional profile (in centimeters) is described for 0≤x≤3 by the graph of y=ln(x+1) above the x–axis. The region under this graph, bounded on the left by x=0 and on the right by x=3, is revolved about the horizontal line y=−2 to form the interior cavity of the basin.
(a) For integration with respect to x, write an exact integral expression whose value equals the volume of the basin’s cavity. Identify the expressions you use for the larger and smaller radii of each representative cross-section.
(b) Solve y=ln(x+1) for x in terms of y and state the corresponding y–values that describe the region.
(c) Using the variables from part (b), write an exact integral with respect to y that also computes the volume.
(d) Without carrying out full analytic integration, explain which of the two integrals is more convenient for numerical approximation and give two specific reasons supporting your choice.