Determine the area of the shaded region in the following figures.

Verified step by step guidance

1. Step 1: Identify the curves that bound the shaded region. The graph shows the shaded region enclosed by the curves y = 2 - x/2 (a straight line) and x = 2y^2 (a parabola).
2. Step 2: Determine the points of intersection of the two curves. To find these points, solve the equations y = 2 - x/2 and x = 2y^2 simultaneously. Substitute x = 2y^2 into y = 2 - x/2 to find the y-values of the intersection points.
3. Step 3: Set up the integral to calculate the area. Since the region is bounded vertically (y-values), express x in terms of y for both curves. The parabola x = 2y^2 represents the left boundary, and the line x = 2 - y/2 represents the right boundary. The area is given by the integral of (right curve - left curve) with respect to y.
4. Step 4: Determine the limits of integration. The limits are the y-values of the intersection points found in Step 2. These limits define the range over which the area is calculated.
5. Step 5: Write the integral for the area. The area is given by the integral A = ∫[y1 to y2] [(2 - y/2) - (2y^2)] dy, where y1 and y2 are the limits of integration. Simplify the integrand and evaluate the integral to find the area.

Key Concepts

Area Between Curves

The area between two curves can be calculated using definite integrals. To find this area, one must identify the upper and lower functions over the interval of interest. The area is given by the integral of the difference between the upper curve and the lower curve, integrated with respect to the variable of interest, typically x or y.

Definite Integrals

Definite integrals represent the accumulation of quantities, such as area under a curve, over a specified interval. The integral is evaluated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The limits of integration correspond to the points where the curves intersect, defining the bounds of the area to be calculated.

Finding Points of Intersection

To determine the area between two curves, it is essential to find their points of intersection. This involves solving the equations of the curves simultaneously to identify the x-values where they meet. These intersection points serve as the limits of integration for calculating the area, ensuring that the correct region is being evaluated.