Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer.
Enclosed by y = x^2 − 4x + 1 and y = −x^2 + 4x − 5
Here is a sketch
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E2+−+4x+%2B+1+and+y+%3D+−x%5E2+%2B+4x+−+5
Easy to find their intersection points:
(1, -2) and (3, -2)
effective height between x = 1 and x = 3
= (-x^2 + 4x - 5) - (x^2 - 4x + 1)
= -2x^2 +8x -6
area = ∫(-2x^2 + 8x - 6) dx from x = 1 to 3
= [(-2/3)x^3 + 4x^2 - 6x] from 1 to 3
= ( -18 + 36 - 18) - (-2/3 + 4 - 6)
= 8/3
check my arithmetic
To find the area of the indicated region enclosed by the curves y = x^2 - 4x + 1 and y = -x^2 + 4x - 5, we can follow these steps:
Step 1: Graph the curves:
To begin, we need to graph the curves y = x^2 - 4x + 1 and y = -x^2 + 4x - 5 on a coordinate plane. This will help us visually understand the region we are interested in.
Step 2: Identify where the curves intersect:
Next, we need to find the x-values where the two curves intersect. To do this, we can set the two equations equal to each other and solve for x:
x^2 - 4x + 1 = -x^2 + 4x - 5
Simplifying the equation, we get:
2x^2 - 8x + 6 = 0
Now, we solve this quadratic equation by factoring or using the quadratic formula. Alternatively, you can use graphing technology or an online graphing calculator to find the x-values where the curves intersect.
Step 3: Find the bounds of integration:
Once we have the x-values where the curves intersect, we can determine the bounds of integration for finding the area. The bounds will be the x-values where the curves intersect.
Step 4: Integrate to find the area:
Finally, we integrate the difference between the two functions with respect to x over the bounds of integration to find the area of the enclosed region:
Area = ∫[lower bound, upper bound] (y1 - y2) dx
where y1 is the upper curve and y2 is the lower curve.
Step 5: Use technology to check your answer:
To verify your result, you can use technology such as graphing calculators or online graphing tools to graph the given curves and find the shaded region. You can also use numerical integration tools to compute the area and compare it to your calculated value.
By following these steps, you should be able to find the area of the region enclosed by the given curves.