Use the limit process to find the area of the region between f (x)=10x+8 and the x-axis on the interval [0,3]
Subdivide the area into n vertical rectangles, each of width
w=(3-0)/n.
The (x-)position of the ith rectangle is x(i)=(i-1)(3/n)=3(i-1)/n.
The height is
h(i)=10x(i)+8
The area is
a(i)=w*h(i)
The total area is therefore A
n
∑ a(i)
i=1
Evaluate the summation and calculate the area A by taking limit n → ∞.
To find the area of the region between the function f(x) = 10x + 8 and the x-axis on the interval [0, 3] using the limit process, we can use the following steps:
1. Divide the interval [0, 3] into equal subintervals. Let's choose n subintervals for this example. Each subinterval will have a width of Δx = Δx_i = (3 - 0) / n.
2. Choose a point xi for each subinterval i. A common choice is xi = (i * Δx), where i represents the index of each subinterval from 0 to n.
3. Now, calculate the height of the function f(xi) for each subinterval. In this case, f(xi) = 10(xi) + 8.
4. Determine the area of each individual rectangle by multiplying the width Δx and the height f(xi). Each rectangle's area will be A_i = f(xi) * Δx.
5. To find the total area, sum up all the individual rectangle areas. We can express this as the sum from i = 1 to n of A_i.
6. Finally, take the limit as n approaches infinity of the sum from i = 1 to n of A_i. This will give us the exact area between the function f(x) and the x-axis on the interval [0, 3].
By calculating this limit, we can find the area using the limit process.