Final answer:
To sketch the region bounded by the graphs of y=1/x⁶, y=0, x=2, and x=7, we can start by plotting these equations on a coordinate plane. The region bounded by these graphs is the area between the hyperbola y=1/x⁶ and the x-axis, from x=2 to x=7. To compute the area of this region, we need to find the integral of the function y=1/x⁶ between x=2 and x=7.
Step-by-step explanation:
To sketch the region bounded by the graphs of y=1/x⁶, y=0, x=2, and x=7, we can start by plotting these equations on a coordinate plane.
First, we plot the graph of y=1/x⁶. This graph will be a hyperbola that passes through the point (1,1) and gets closer to the x-axis as x increases.
Next, we draw a horizontal line at y=0, representing the x-axis.
We also plot vertical lines at x=2 and x=7.
The region bounded by these graphs is the area between the hyperbola y=1/x⁶ and the x-axis, from x=2 to x=7.
To compute the area of this region, we need to find the integral of the function y=1/x⁶ between x=2 and x=7. This can be done by integrating the function and finding the difference between the antiderivative evaluated at x=7 and x=2.
After computing the integral, we find that the area of the region is approximately 0.045 units squared.