For the function f(x)=10-4(x^2), find a formula for the lower sum obtained by dividing the interval [0,1] into n equal subintervals. Then take the limit as n->infinity to calculate the area under the curve over [0,1].
I only need help with the first part. I don't really understand how to find the formula for the lower sum.
i wrote down in my notes that a=w*h
w= (upper limit-lower limit)/n
h= (i*w)^2
and a= h[(n(n+1)(2n+1))/6]
but I don't really understand what "i" stands for, so I don't know how to use the formula.
YA i can't fingure out how to use it either im thinking that it could be the xi = a+x(change of x)
To find a formula for the lower sum obtained by dividing the interval [0,1] into n equal subintervals, you are on the right track with your notes. Let me explain how to use the formula correctly.
First, let's clarify what each variable represents in the formula:
- n: Number of subintervals into which the interval [0,1] is divided.
- a: Area under the curve (desired result).
- w: Width of each subinterval.
- h: Height of each rectangle.
- i: Index of each subinterval, ranging from 1 to n.
Now, let's go step by step to calculate the lower sum:
1. Determine the width of each subinterval (w):
The interval [0,1] needs to be divided into n equal subintervals. To calculate the width of each subinterval, you divide the length of the interval by the number of subintervals:
w = (1 - 0) / n = 1/n
2. Calculate the height of each rectangle (h):
The height of each rectangle depends on the function values at specific points within each subinterval. In this case, we want to find the lower sum, so we choose the smallest value within each subinterval.
Let's denote the left endpoint of each subinterval by xi. Since the interval [0,1] is divided into n equal subintervals, the xi values would be:
xi = 0, 1/n, 2/n, 3/n, ..., (n-1)/n
Now, substitute these xi values into the function f(x) = 10 - 4(x^2) to calculate the corresponding function values. Find the smallest value among the function values in each subinterval and use it as the height (h) of the rectangle.
3. Calculate the area of each rectangle (a):
Once you have the width (w) and height (h) of each rectangle, you can calculate the area of each rectangle using the formula a = w * h. Multiply the width and height for each subinterval and note down these individual rectangle areas.
4. Sum up the areas to find the lower sum (A):
The lower sum (A) is obtained by adding up all the individual rectangle areas. As you correctly noted, the formula for the sum of areas is a = h[(n(n+1)(2n+1))/6], where h represents the height and n is the number of subintervals. Substitute the individual rectangle areas that you calculated in step 3 into the formula to find A, the approximate area under the curve.
Once you have the formula for the lower sum (A), you can take the limit as n approaches infinity to calculate the exact area under the curve over the interval [0,1].