Question

Find the area of the region described in the following exercises.The region bounded by y=2−|x|and y=$$x^2$$

Accepted Answer

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. 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 \u003c 0, solve 2 + x = $$x^2. $$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. 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. 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.

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

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

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Concetti chiave

Area Between Curves

Finding Intersection Points

Integration

Find the area of the region described in the following exercises.The region bounded by y=2−|x|and y=x^2