Question

Find the area of the shaded region. (We suggest that you use technology to check your answers.)

The first curve, labeled y = x ^2 − 2x + 1

The second curve, labeled y = 5 − x^2


The graph is marked 0-2 and then 2-3. It wants you to find the area between the lines at these intervals if that makes sense? There is two because the lines cross. My teacher did not give us an example like this in our notes.

Answer 1

The area of the shaded region between the curves
`y = x^2 - 2x + 1`and
`y = 5 - x^2,` within the intervals 0 to 2 and 2 to 3, is 6.33 square units.

Explanation:

To find the area between the curves within the given intervals, integrate the absolute difference of the functions over each interval and then sum these integrals. First, identify the points of intersection between the curves. Set the equations equal to each other:
`x^2 - 2x + 1 = 5 - x^2.` Solving this yields x = ±2.

Between 0 and 2, the upper curve is
`y = 5 - x^2`, and the lower curve is
`y = x^2 - 2x + 1.` Calculate the definite integral of
`|(5 - x^2) - (x^2 - 2x + 1)|`from 0 to 2. This yields an area of 6 units^2 for this interval.

Between 2 and 3, the upper curve remains
`y = 5 - x^2`, but the lower curve changes to
`y = x^2 - 2x + 1.` Calculate the definite integral of
`|(5 - x^2) - (x^2 - 2x + 1)|` from 2 to 3. This results in an area of approximately 0.33 units^2 for this interval.

Summing the areas of the two intervals gives the total shaded region area of 6.33 square units. This demonstrates how to approach finding areas between curves by calculating the absolute difference of the functions and integrating within the specified intervals, considering the changing upper and lower curves over those intervals.

Answer 2

To find the area of the shaded region between the curves y = x^2 - 2x + 1 and y = 5 - x^2, one must determine the intersection points and then calculate the areas using integrals over the intervals from 0 to 2 and from 2 to 3, summing the results.

Step-by-step explanation:

The student is asked to find the area of the shaded region between the curves y = x^2 - 2x + 1 and y = 5 - x^2. The integral calculus can be used to calculate the area between two curves by integrating the difference of the functions over the given interval. This problem requires the evaluation of two separate integrals, as the curves intersect and create two distinct regions to be calculated.

Since there's a point of intersection, we need to find the x-values where the curves meet, which requires setting the equations equal to each other to solve for x. Subsequently, we would perform the integral of the difference between the upper and lower functions over the relevant x-intervals (0 to 2 and 2 to 3).

To provide a step-by-step solution, first solve for the intersections:

1. Set x^2 - 2x + 1 equal to 5 - x^2 and solve for x to find the intersection points.
2. Calculate two separate integrals, one from the leftmost intersection to the point where curves switch (0 to 2) and the other from that switching point to the rightmost intersection (2 to 3).
3. Find the areas by integrating the upper curve minus the lower curve for each interval and add them together to obtain the total shaded area.

Note that for the interval from 0 to 2, the upper function is 5 - x^2 and the lower is x^2 - 2x + 1, and for 2 to 3, it switches.