The region R looks like this: ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_graphik_30.png)
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 = x2 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 = x2 to the graph , and one piece where the horizontal slices go from the graph y = x2 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 = x2 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 = x2 with coordinates to a point on the graph of with coordinates (x,y) = (y2, y). The length of this slice is , so its area is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_73.png)
Writing the integral with the limits of integration we figured out earlier, the area of this piece of the region is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_74.png)
For the upper piece of R, the region goes from y = 2 to y = 42 = 16. A slice extends from the graph of y = x2 to the line x = 4. The width of a slice is , so the area is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_76.png)
This means the area of this piece of the region is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_77.png)
Now that we have two areas, we just need to add them together to get the total area of the region R: ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_area_latek_78.png)
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. |