MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A function f is defined by f(x)=x+26−1 for x≥0. Let R be the finite region enclosed between the graph of f, the x-axis, and the vertical line x=6.
(a) Determine the x-coordinate where the graph of f intersects the x-axis for x≥0, and justify that this point lies between x=0 and x=6.
(b) Write, but do not evaluate, a definite integral that represents the area of region R.
(c) Evaluate the integral from part (b) to find the exact area of R.
(d) Suppose region R is revolved about the line y=−1 rather than the x-axis. Write a new integral for the resulting volume and evaluate to obtain the exact value.