Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval.
y = −4x + 6, [0, 1]
To find the area between the graph of the function and the x-axis over the given interval using the limit process, you can follow these steps:
1. Start by setting up the integral. Since we're finding the area between the graph and the x-axis, we need to integrate the absolute value of the function.
So, the integral setup would be:
Area = ∫[a,b] |f(x)| dx
In this case, the function is y = -4x + 6, and the interval is [0, 1]. So, our integral becomes:
Area = ∫[0,1] |-4x + 6| dx
2. Next, we need to find the limits of integration. Since the interval is [0,1], the lower limit of integration is a = 0, and the upper limit is b = 1.
3. Now, we need to break the interval [0,1] into small subintervals of equal width. We can use a small value of Δx to represent the width of each subinterval.
4. The next step is to evaluate the function at each point in the subinterval. In this case, we need to evaluate |-4x + 6| for each x value within the subinterval.
5. After evaluating the function at each point, we multiply the function value by the width of the subinterval Δx to get the area of each individual rectangle.
6. Finally, we add up all the areas of the rectangles to get an approximation of the area between the graph and the x-axis.
As the value of Δx approaches zero, the sum of the areas of the rectangles becomes a better approximation of the actual area between the graph and the x-axis.
This process is better represented using a definite integral, which allows us to find the exact value of the area. However, for the purpose of explaining the limit process, we use this approximation method.