How do you find the area of the region between the curves #y=x-1# and #y^2=2x+6# ?

Question

Charles Arocha · Accepted Answer

The area of the region between #y=x-1# and #y^2=2x+6# is 18.

Let us look at some details.

The region looks like this:

William Abdalla · Answer

undefined To find the area of the region between the curves $ y = x - 1 $ and $ y^2 = 2x + 6 $, you need to first determine the points of intersection of these two curves. Then, you integrate the absolute difference between the functions from the leftmost intersection point to the rightmost intersection point. The formula for the area between curves $ f(x) $ and $ g(x) $ from $ x = a $ to $ x = b $ is:

$$ \text{Area} = \int_{a}^{b} |f(x) - g(x)| \, dx $$

So, you find the points of intersection, set up the integral with the absolute difference of the two functions, and then integrate with respect to $ x $ from the leftmost to the rightmost intersection points.

Charles Anctil · Answer

undefined To find the area of the region between the curves $y = x - 1$ and $y^2 = 2x + 6$, you first need to determine the points of intersection of the two curves by solving the system of equations formed by them. Then, integrate the difference of the upper curve and the lower curve with respect to $x$ over the interval where they intersect. This will give you the area of the region between the curves.