Questions: Sketch the region enclosed by x+y^2=20 and x+y=0. Decide whether to integrate with respect to x or y. Then find the area of the region.

Transcript text: Sketch the region enclosed by $x+y^{2}=20$ and $x+y=0$. Decide whether to integrate with respect to $x$ or $y$. Then find the area of the region.

Solution

Solution Steps

To solve this problem, we need to find the area of the region enclosed by the curves $ x + y^2 = 20 $ and $ x + y = 0 $. We will first find the points of intersection of these curves. Then, we will decide whether to integrate with respect to $ x $ or $ y $ based on the simpler integral setup. Finally, we will compute the area using definite integrals.

Find the points of intersection by solving the system of equations $ x + y^2 = 20 $ and $ x + y = 0 $.
Determine the bounds of integration.
Set up the integral to find the area of the region.
Compute the integral.

Step 1: Find the Points of Intersection

To find the points of intersection of the curves defined by the equations $ x + y^2 = 20 $ and $ x + y = 0 $, we solve the system of equations. The solutions yield the points of intersection as $ (-5, 5) $ and $ (4, -4) $.

Step 2: Determine the Bounds of Integration

From the intersection points, we identify the $ y $-values as $ y = 5 $ and $ y = -4 $. Thus, the bounds of integration for $ y $ are from $ y = -4 $ to $ y = 5 $.

Step 3: Set Up the Integral

We express $ x $ in terms of $ y $ for both curves:

From $ x + y^2 = 20 $, we have $ x = 20 - y^2 $.
From $ x + y = 0 $, we have $ x = -y $.

The area $ A $ of the region enclosed by the curves can be calculated using the integral: \[ A = \int_{-4}^{5} \left( (20 - y^2) - (-y) \right) \, dy = \int_{-4}^{5} (20 - y^2 + y) \, dy \]

Step 4: Compute the Area

Evaluating the integral gives us the area of the region: \[ A = \frac{243}{2} = 121.5 \]