Find the area of the region described in the following exercises.


The region bounded by y=2−|x|and y=x^2

Verified step by step guidance

1. Step 1: Identify the curves that bound the region. The region is bounded by y = 2 - |x| and y = x^2. The first curve, y = 2 - |x|, is a V-shaped function symmetric about the y-axis, while the second curve, y = x^2, is a parabola opening upwards.
2. Step 2: Determine the points of intersection between the two curves. To find these points, solve the equation 2 - |x| = x^2. Break this into two cases: (a) when x ≥ 0, solve 2 - x = x^2, and (b) when x < 0, solve 2 + x = x^2.
3. Step 3: Set up the integral to calculate the area. The region is symmetric about the y-axis, so you can calculate the area for x ≥ 0 and then double it. For x ≥ 0, the area is given by the integral of the top curve (y = 2 - x) minus the bottom curve (y = x^2) over the interval determined by the intersection points.
4. Step 4: Write the integral expression for the area. For x ≥ 0, the area is: \( \int_{x_1}^{x_2} \left( (2 - x) - x^2 \right) dx \), where x_1 and x_2 are the intersection points found in Step 2. Double this result to account for the symmetry.
5. Step 5: Evaluate the integral. Break the integral into simpler parts: \( \int_{x_1}^{x_2} 2 dx \), \( \int_{x_1}^{x_2} x dx \), and \( \int_{x_1}^{x_2} x^2 dx \). Combine these results to find the total area, ensuring you double the result for symmetry.

Key Concepts

Area Between Curves

The area between two curves can be found by integrating the difference of the functions that define the upper and lower boundaries over a specified interval. In this case, the area is determined by the vertical distance between the curves y = 2 - |x| and y = x^2, integrated across the x-values where they intersect.

Finding Intersection Points

To calculate the area between two curves, it is essential to find their points of intersection. This involves setting the equations equal to each other and solving for x. The intersection points will define the limits of integration for calculating the area.

Integration

Integration is a fundamental concept in calculus used to calculate areas under curves. In this context, it involves computing the definite integral of the difference between the two functions over the interval defined by their intersection points, yielding the total area of the region bounded by the curves.