use the limit process to find the area of the region between f(x) = x^2 + 2 a interval [0, 3]
I think all you do is integrate this from 0 to 3
So the integral of x^2 + 2:
integral of x^2:
x^3/3
integral of 2:
2x
so integral:
x^3/3 + 2x
now integrate this from 0 to 3:
from 3:
3^3/3 + 2*3 = 15
from 0:
0^2+2*0 = 0
15 - 0 = 15
I think this is how you approach this. I don't know what the limit process is, but I can't see what else you could do considering that its given you intervals to integrate at.
To find the area of the region between the function f(x) = x^2 + 2 over the interval [0, 3] using the limit process, we can break the interval into small sub-intervals and approximate the area of each sub-interval using rectangles.
Here's how you can do it step by step:
Step 1: Divide the interval [0, 3] into n equal sub-intervals. Let's call the width of each sub-interval Δx, which is given by Δx = (b - a)/n, where a and b are the endpoints of the interval.
In our case, a = 0 and b = 3, so the width of each sub-interval Δx = (3 - 0)/n = 3/n.
Step 2: Choose a representative point in each sub-interval. Let's call this point xi, which is xi = a + iΔx, where i is the index of the sub-interval (i = 0, 1, 2, ..., n-1).
In our case, xi = 0 + i(3/n) = 3i/n.
Step 3: Find the value of f(xi) for each representative point xi. In our case, f(xi) = (3i/n)^2 + 2.
Step 4: Approximate the area of each sub-interval. Multiply the width Δx by f(xi) to get the area of each rectangle. In our case, the area of each rectangle is ΔA = (3/n) * [(3i/n)^2 + 2].
Step 5: Sum up the areas of all the rectangles. This can be done using a summation notation. Add up all the areas of the rectangles to get the approximate area under the curve. In our case, the sum will be:
Approximate area = Σ[(3/n) * [(3i/n)^2 + 2]].
The Σ represents the summation notation. The limits of summation are from i = 0 to i = n-1.
Step 6: Take the limit as n approaches infinity. This will give us the exact area under the curve. You can rewrite the summation using sigma notation:
Exact area = lim(n→∞) Σ [(3/n) * [(3i/n)^2 + 2]].
Step 7: Simplify and evaluate the limit. You will end up with an expression involving n, which you can simplify to obtain the exact area.
Keep in mind that as n approaches infinity, the sub-intervals become infinitesimally small, giving us a more accurate approximation of the area.
So, by following these steps, you can use the limit process to find the area of the region between f(x) = x^2 + 2 over the interval [0, 3].