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 given curves are x = (y - 2)^2 / 3 and y = 8 - x. The shaded region lies between these two curves.
2. Step 2: Determine the points of intersection of the two curves. To find these points, solve the equations simultaneously. Substitute x = (y - 2)^2 / 3 into y = 8 - x to find the y-values where the curves intersect.
3. Step 3: Set up the integral to calculate the area. Since the region is bounded vertically (y-values), express the area as an integral with respect to y. The limits of integration will be the y-values found in Step 2.
4. Step 4: Write the integral for the area. The area is given by the integral of the difference between the rightmost curve (x = (y - 2)^2 / 3) and the leftmost curve (x = 8 - y) with respect to y. The integral is: ∫[(y - 2)^2 / 3 - (8 - y)] dy, with limits determined in Step 2.
5. Step 5: Simplify the integrand and compute the integral. Expand and combine terms inside the integral, then integrate term by term. Evaluate the definite integral using the limits of integration to find the area of the shaded region.

Key Concepts

Area Between Curves

The area between curves is calculated by integrating the difference between the upper and lower functions over a specified interval. In this case, the area of the shaded region is found by determining the points of intersection of the curves and integrating the top curve minus the bottom curve within that interval.

Integration

Integration is a fundamental concept in calculus that allows us to find the area under a curve. It involves calculating the integral of a function, which represents the accumulation of quantities, such as area. In this problem, definite integrals will be used to compute the area of the shaded region between the two curves.

Finding Points of Intersection

Finding points of intersection involves solving the equations of the curves to determine where they meet. These points are crucial for setting the limits of integration when calculating the area between the curves. In this case, solving the equations x = (y - 2)²/3 and y = 8 - x will yield the necessary intersection points.