Find the area of the region described in the following exercises.


The region bounded by y=x^2−2x+1 and y=5x−9

Verified step by step guidance

1. Step 1: Identify the curves that bound the region. The first curve is y = x^2 - 2x + 1, which is a parabola, and the second curve is y = 5x - 9, which is a straight line.
2. Step 2: Find the points of intersection between the two curves by setting their equations equal to each other: x^2 - 2x + 1 = 5x - 9. Rearrange the equation to form a standard quadratic equation: x^2 - 7x + 10 = 0.
3. Step 3: Solve the quadratic equation x^2 - 7x + 10 = 0 using factoring, the quadratic formula, or another method. The solutions will give the x-coordinates of the points of intersection.
4. Step 4: Set up the integral to calculate the area between the curves. The area is given by the integral of the difference between the upper curve (y = 5x - 9) and the lower curve (y = x^2 - 2x + 1) over the interval determined by the points of intersection. The integral is: ∫[a, b] ((5x - 9) - (x^2 - 2x + 1)) dx, where 'a' and 'b' are the x-coordinates of the intersection points.
5. Step 5: Simplify the integrand ((5x - 9) - (x^2 - 2x + 1)) to get -x^2 + 7x - 10. Then, compute the integral ∫[a, b] (-x^2 + 7x - 10) dx by finding the antiderivative and evaluating it at the bounds 'a' and 'b'.

Key Concepts

Finding the Area Between Curves

To find the area between two curves, we first identify the points of intersection, which serve as the limits of integration. The area is then calculated by integrating the difference between the upper curve and the lower curve over the interval defined by these points. This process involves setting up the integral of the function representing the upper curve minus the function representing the lower curve.

Identifying Functions and Their Intersections

In this problem, we have two functions: a quadratic function, y = x^2 - 2x + 1, and a linear function, y = 5x - 9. To find the area between them, we need to determine where these two functions intersect. This is done by setting them equal to each other and solving for x, which gives us the points that will be used as limits for integration.

Integration Techniques

Integration is a fundamental concept in calculus used to calculate areas under curves. In this context, we will use definite integration to find the area between the two curves. The integral of a function can be computed using various techniques, such as substitution or integration by parts, depending on the complexity of the functions involved.