The region *R* looks like this: Since the problem says to use horizontal slices, life will be a little bit difficult. We could hit our heads on the wall and hope it works itself out, but it won't. We might as well dive in head first. A slice at the bottom of the region has its left endpoint on the line *y* = *x*^{2} and its right endpoint on the line . Meanwhile, a slice at the top of the region has its right endpoint on the line *x* = 4. Don't throw your pencil at a wall. It's easier to handle than you may think. We have to split the region into two pieces: one piece where the horizontal slices go from the graph *y* = *x*^{2} to the graph , and one piece where the horizontal slices go from the graph *y* = *x*^{2} to the line *x* = 4. Let's look at the lower piece first, and start by figuring out the limits of integration. We know that *y* = *x*^{2} and intersect at (1,1), so *y* = 1 will be our lower limit of integration. The upper limit of integration is where the graph intersects the line *x* = 4, which is at (4,2). The upper limit of integration is *y* = 2. The slice at position *y* goes from a point on the graph of *y* = *x*^{2} with coordinates to a point on the graph of with coordinates (*x*,*y*) = (*y*^{2}, *y*). The length of this slice is , so its area is Writing the integral with the limits of integration we figured out earlier, the area of this piece of the region is For the upper piece of *R*, the region goes from *y* = 2 to *y* = 4^{2} = 16. A slice extends from the graph of *y* = *x*^{2} to the line *x* = 4. The width of a slice is , so the area is This means the area of this piece of the region is Now that we have two areas, we just need to add them together to get the total area of the region *R*: That was quite a bit of work. It would have been faster to use vertical slices, but the evil Shmoop question writing demon felt a little torture was in order. Despite our growling stomachs from all the talk about fish sticks, we solved it. |