Question

In Exercises \(15-34,\) find the area of the regions enclosed by the lines and curves. $$y=\sqrt{|x|}$$ and $$5 y=x+6$$

Short answer

The area enclosed by the lines \(y=\sqrt{|x|}\) and \(5y=x+6\) is equal to 9.8 unit square.

Step-by-step solution

Find the intersection points

To find where the curves intersect, set \(y=\sqrt{|x|}\) equal to \(5y=x+6\). Solving this equation gives two points of intersection, \((-1,1)\) and \((6, 5/2)\).

Setup the integral

Since we have the points of intersection, we can now set up the integral to calculate the area. The area A between the curves from \(x=a\) to \(x=b\) is given by: \(A=\int_{a}^{b} [f(x) - g(x)] dx\). Here, \(f(x)= \frac{x+6}{5}\) and \(g(x)= \sqrt{|x|}\). We will take two cases, considering absolute value, from \(-1\) to \(0\) and \(0\) to \(6\).

Evaluate the integral

Now, we need to evaluate the integral. For \(x\) from \(-1\) to \(0\), the area equals \(\int_{-1}^{0}[\frac{x+6}{5} – (-\sqrt{x})] dx = \frac{4}{5}\). And for \(x\) from \(0\) to \(6\), the area is \(\int_{0}^{6}[\frac{x+6}{5} - \sqrt{x}] dx = \frac{17}{2}\).

Sum up the areas

The total area enclosed by the curves is the sum of the two areas computed in step 3, that is, Total Area = \(\frac{4}{5} + \frac{17}{2} = 9.8\) unit square.