Sketch the given region R
and then find its area.
R is the region bounded by the curve y=1/x^2 and the lines y=x and y=x/8.
Great. No matter which way you do it, you have to divide the region into two parts because of the boundary changes. So, the area could be either
∫[0,1] (x - x/8) dx + ∫[1,2] (1/x^2 - x/8) dx = 3/4
∫[0,1/4] (8y - y) dy + ∫[1/4,1] (1/√y - y) dy = 3/4
To sketch the region R, we can start by plotting the given curves and lines on a coordinate plane.
First, let's plot the curve y=1/x^2. For this curve, we can choose some x-values, calculate the corresponding y-values, and plot the points. Here are a few points that lie on the curve:
(x, y)
1, 1
2, 1/4
3, 1/9
4, 1/16
-1, 1
-2, 1/4
-3, 1/9
-4, 1/16
Plotting these points will give us a curve that starts from the top right, curves downwards, and approaches the x-axis.
Next, let's plot the line y=x. This line represents a diagonal that passes through the origin (0,0) with a slope of 1. Therefore, we can easily draw this line.
Finally, let's plot the line y=x/8. This line represents a diagonal that also passes through the origin but has a smaller slope of 1/8. So, the line is not as steep as y=x.
After plotting all three curves and lines mentioned above, the region R will be the region that is bounded by the curve y=1/x^2, the line y=x, and the line y=x/8.
To find the area of region R, we need to integrate the appropriate function. In this case, since we are dealing with curves and lines, we can integrate with respect to x.
To find the limits of the integral, we need to find the x-values at which the curves intersect each other. In this case, the actual limits are x = -4 and x = 4.
Using the above information, we can set up the integral to find the area:
A = ∫[from -4 to 4] (1/x^2 - x/8 - x) dx
Simplifying this integral will give us the area of the region R.