Let R be the region bounded by the curves $y = e^x$ and $y = 1 + (e − 1)x$.
Write down an integral that represent the volume of the solid $S$ obtained by rotating $R$ about the $x$-axis
When I solve this problem, 
I get a negative volume. Why am I getting a negative volume?
If you plot out the graph, you will see that for the bounded region, the linear line is above the exponential function. You have used a smaller quantity to subtract a bigger quantity, hence, you get a negative number.