What is the area of the region in the xy-plane bounded by the graphs of x=1995, x=1996, y=1997, and y=1998?
well, it's a square with side=1
. . .
To find the area of the region in the xy-plane bounded by the graphs of x=1995, x=1996, y=1997, and y=1998, we need to determine the limits of integration and set up the integral to calculate the area.
The region is a rectangle with x-coordinates between 1995 and 1996, and y-coordinates between 1997 and 1998.
The area can be calculated by integrating the constant function 1 over this region. Let's set up the integral as follows:
∫∫1 dA,
where the limits of integration are x=1995 to x=1996, and y=1997 to y=1998.
Integrating with respect to x first, we get:
∫[1995,1996] ∫[1997,1998] 1 dy dx.
Integrating with respect to y, we get:
∫[1995,1996] [y] [1997,1998] dx.
Now, plugging in the limits of integration, we have:
∫[1995,1996] [1998-1997] dx.
Simplifying further, we get:
∫[1995,1996] dx.
Evaluating the integral, we have:
[x] [1995,1996].
Substituting the limits of integration, we get:
[1996]-[1995].
Calculating further, we have:
1996-1995.
Thus, the area of the region bounded by the graphs of x=1995, x=1996, y=1997, and y=1998 is 1 square unit.
To find the area of the region bounded by the graphs of x=1995, x=1996, y=1997, and y=1998 in the xy-plane, we first need to visualize the region.
The graph of x=1995 is a vertical line passing through the x-coordinate 1995. Similarly, x=1996 is a vertical line passing through the x-coordinate 1996. These two lines define the left and right boundaries of the region.
The graph of y=1997 is a horizontal line passing through the y-coordinate 1997. Similarly, y=1998 is a horizontal line passing through the y-coordinate 1998. These two lines define the bottom and top boundaries of the region.
Now, we can see that the region is a rectangle with vertical sides of length 1 (1996 - 1995) and horizontal sides of length 1 (1998 - 1997).
To find the area of a rectangle, we multiply its length by its width. In this case, the length is 1 and the width is also 1.
Therefore, the area of the region bounded by the graphs of x=1995, x=1996, y=1997, and y=1998 is 1 square unit.