Math Problem Statement
Find the area of the region enclosed by the graph y=x^2-4x+3 and y=0 between x=0 and x=4.
Solution
We are tasked with finding the area of the region enclosed by the graph and the line between and .
Step 1: Find the points where the curve intersects the x-axis
To determine the region enclosed by the graph and the x-axis, we need to find the points where the curve intersects the x-axis (i.e., where ). Set the equation and solve for :
We can factor the quadratic equation:
Thus, the roots are:
Step 2: Set up the integral
Now, we need to find the area between the curve and the x-axis from to . The area is given by the integral of the function over this interval:
Step 3: Compute the integral
Now, let's compute the integral:
We will now evaluate this from to :
First, substitute :
Next, substitute :
Thus, the area is:
Step 4: Interpret the result
The area of the region enclosed by the graph and the x-axis between and is .
Would you like more details on any part of the process? Let me know if you have further questions!
Here are some related questions you might find interesting:
- How do you find the area between two curves?
- What is the significance of the roots in finding the enclosed area?
- Can you apply this method to other polynomial functions?
- What happens if the function does not intersect the x-axis within the given interval?
- How do you handle cases where the graph is below the x-axis in finding the area?
Tip: Always ensure that the function is properly bounded by the region of interest before setting up your integral!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Definite Integration
Area under a Curve
Formulas
Quadratic equation formula ax^2 + bx + c = 0
Definite integral formula ∫[a,b] f(x) dx
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 10-12