Application Of Integrals

Step 1: Find the points of intersection by setting the equations equal: . These are the limits of integration.
Step 2: Set up the definite integral for the area. The area is the integral of the upper curve minus the lower curve. Here, is the upper curve and is the lower curve. Step 3: Use the property of even functions to simplify the integral. Since is an even function, . Step 4: Evaluate the integral.

Step 1: Visualize the region.
The parabola opens to the right with its vertex at the origin. The line is a vertical line. The region is bounded on the left by the y-axis ( ) and on the right by . The region is symmetric about the x-axis.
Step 2: Set up the integral.
We can find the area of the top half and double it. From , the upper half is . The limits of integration are from to . Step 3: Evaluate the integral.
Using the power rule for integration: The area is square units.