Question

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.y = x,  y = 1/x²,  x = 2

Accepted Answer

First, identify the region enclosed by the curves $$y = x$$, $$y = \frac{1}{x^2$}$$, and the vertical line $$x = 2. To do $$this, find the points of intersection between $$y = x$$ and $$y = \frac{1}{x^2$} by $$solving the equation $$x = \frac{1}{x^2$}. $$Solve $$x = \frac{1}{x^2$} by $$multiplying both sides by $$x^2 to $$get $$x^3 = 1$. Then find the real root(s) of this equation to determine the x$-coordinate(s) of the intersection point(s). Determine which curve is on top and which is on the bottom between the points of intersection and up to x = 2$. This will help set up the integral for the area between the curves correctly. Set up the definite integral for the area of the enclosed region as $$\int_{a}^{2} \left( 	ext{top function} - 	ext{bottom function} ight) \, dx$$, where a$ is the x$-value of the intersection point found earlier. Evaluate the integral by integrating the difference of the functions y = x$ and y$$ = \frac{1}{$$x^2$$}$$ over the interval $$[a, 2]$$ to find the area of the enclosed region.

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = x, y = 1/x², x = 2

Verified video answer for a similar problem:

Key Concepts

Finding Points of Intersection

Setting Up Definite Integrals for Area

Handling Vertical Boundaries in Integration