Estimate the area under the curve #f(x) = x^2# over the interval #[0,10]# with #5# strips using Left Riemann Sums?

Question

Estimate the area under the curve #f(x) = x^2# over the interval #[0,10]# with #5# strips using  Left Riemann Sums?

Evelyn Agard · Accepted Answer

Since you are using 5 sub intervals, your intervals would be divided like this:
#[0,2], [2,4], [4,6], [6,8], [8,10]# The left riemann sum requires you to use the lower number in every interval listed above as your #x# term, and then evaluate the functional value at that particular #x# value, like this: Note that #2# would be the width of every rectangle formed, since the intervals are of equal width. #1. Area_1 = f(0)*2 = 0^2*2 = 0# #2. Area_2 = f(2)*2 = 2^2*2 = 8# #3. Area_3 = f(4)*2 = 4^2*2 = 32# #4. Area_4 = f(6)*2 = 6^2*2 = 72# #5. Area_5 = f(8)*2 = 8^2*2 = 128# Upon finding the individual areas of each of the five rectangles formed by the five sub-intervals evaluated at the "left side" #x# value, all that is left to do is add all the individual areas up: #Total Area = 0 + 8 + 32 + 72 + 128 = 240# Alternatively, you could use the fact that all the rectangles are the same width, keeping #Deltax# the same:
#Total Sum = 2*(f(0) + f(2) + f(4) + f(6) + f(8))#
#Total Sum = 2*(0 + 4 + 16 + 36 + 64) = 2*(120) = 240# Therefore your left riemann sum would be #240# for the graph #y = x^2#on the interval #[0,10]#

Aurora Alvarado · Answer

# LRS = 240 #

We have:

# f(x) = x^2 #

We want to calculate over the interval #[0,10]# with #5# strips; thus:

# Deltax = (10-0)/5 = 2#

Note that we have a fixed interval (strictly speaking a Riemann sum can have a varying sized partition width). The values of the function are tabulated as follows;

Left Riemann Sum

# LRS = sum_(r=0)^4 f(x_i) \ Deltax_i #
# " " = 2 * (0 + 4 + 16 + 36 + 64) #
# " " = 2 * (120) #
# " " = 240 #

Actual Value

For comparison of accuracy:

# Area = int_0^10 \ x^2 \ dx #
# " " = [x^3/3]_0^10 #
# " " = 1/3[x^3]_0^10 #
# " " = 1/3(1000) #
# " " = 1000/3 #
# " " = 333.3333... #

Ezekiel Alexander · Answer

undefined To estimate the area under the curve $ f(x) = x^2 $ over the interval $[0,10]$ with 5 strips using Left Riemann Sums, follow these steps:

1. Determine the width of each strip:
$$ \Delta x = \frac{b - a}{n} = \frac{10 - 0}{5} = 2 $$

2. Calculate the left endpoints of each strip:
$$ x_0 = 0, \ x_1 = 2, \ x_2 = 4, \ x_3 = 6, \ x_4 = 8, \ x_5 = 10 $$

3. Evaluate the function at each left endpoint:
$$ f(x_0) = 0, \ f(x_1) = 2^2 = 4, \ f(x_2) = 4^2 = 16, \ f(x_3) = 6^2 = 36, \ f(x_4) = 8^2 = 64, \ f(x_5) = 10^2 = 100 $$

4. Calculate the area of each strip:
$$ A_1 = f(x_0) \Delta x = 0 \times 2 = 0 $$
$$ A_2 = f(x_1) \Delta x = 4 \times 2 = 8 $$
$$ A_3 = f(x_2) \Delta x = 16 \times 2 = 32 $$
$$ A_4 = f(x_3) \Delta x = 36 \times 2 = 72 $$
$$ A_5 = f(x_4) \Delta x = 64 \times 2 = 128 $$

5. Sum up the areas of all strips:
$$ \text{Total area} = A_1 + A_2 + A_3 + A_4 + A_5 = 0 + 8 + 32 + 72 + 128 = 240 $$

Therefore, the estimated area under the curve $ f(x) = x^2 $ over the interval $[0,10]$ with 5 strips using Left Riemann Sums is $ 240 $ square units.