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.
Between y = x2 − 4x + 1 and y = −x2 + 4x − 5 for x in [0, 3]
The curves intersect at x=1,3. See
http://www.wolframalpha.com/input/?i=plot+x^2-4x%2B1+and+-x^2%2B4x-5
So, integrate (-x^2+4x-5)-(x^2-4x+1) from 1 to 3.
straightforward power rule stuff.
To find the area of the indicated region between the curves y = x^2 - 4x + 1 and y = -x^2 + 4x - 5 for x in the interval [0, 3], we need to calculate the definite integral of the difference between the upper and lower functions.
First, let's graph the two curves and visually confirm the region we're interested in. You can use any graphing tool or software to plot the functions y = x^2 - 4x + 1 and y = -x^2 + 4x - 5.
Once you have the graphs, you can see that the two curves intersect at two points: (1, -2) and (3, -2). To find the area between them, we need to determine the boundaries of integration.
From the graph, we see that the upper curve is y = x^2 - 4x + 1 and the lower curve is y = -x^2 + 4x - 5. To locate the x-values where the curves intersect, we set the two equations equal to each other:
x^2 - 4x + 1 = -x^2 + 4x - 5
Simplifying the equation, we get:
2x^2 - 8x + 6 = 0
Dividing the equation by 2, we obtain:
x^2 - 4x + 3 = 0
Factoring the quadratic equation, we have:
(x - 1)(x - 3) = 0
So x = 1 or x = 3.
Since the interval for x is defined as [0, 3], we need to integrate in two parts: from 0 to 1 and from 1 to 3.
The area between the curves is given by the integral:
∫[0,3] (upper curve - lower curve) dx = ∫[0,1] [(x^2 - 4x + 1) - (-x^2 + 4x - 5)] dx + ∫[1,3] [(x^2 - 4x + 1) - (-x^2 + 4x - 5)] dx
To calculate this integral, we can simplify the expression inside the integral and then evaluate the integral.
After integrating both parts, we can add up the results from the two separate integrals to find the total area between the two curves on the interval [0, 3].
If you prefer to use technology to check your answer, you can use calculators or online graphing tools that provide options for integrating functions to find the area.