How do you estimate the area under the graph of #f(x)=4sqrt(x)# from #x=0# to #x=4# using four approximating rectangles and right endpoints?

Question

Ezra Adams · Accepted Answer

Find the value of #f(x)# for each integer value: #0, 1, 2, 3, 4#
giving 4 sub-ranges:
#[f(0),f(1)#] # = [0,2.0#],
#[f(1),f(2)#] #= [2.0, 5.6#] (approx.)
#[f(2),f(3)#] # = [5.6, 6.8#] (approx.)
#[f(3),f(4)#] # = [6.8, 8.0#]

Use the mid point of each range as the rectangular height for the #4# rectangles (note: the width of each rectangle is 1)

Approximate area
# = 1.0 + 3.8 + 6.2 + 7.4#
# = 18.4#

Ezekiel Alexander · Answer

undefined To estimate the area under the graph of $ f(x) = 4\sqrt{x} $ from $ x = 0 $ to $ x = 4 $ using four approximating rectangles and right endpoints, follow these steps:

1. Divide the interval $ [0, 4] $ into four equal subintervals. Since we have four rectangles, each subinterval will have a width of $ \Delta x = \frac{4}{4} = 1 $.

2. Choose the right endpoint of each subinterval as the height of the corresponding rectangle. Calculate the height of each rectangle by evaluating the function $ f(x) = 4\sqrt{x} $ at the right endpoint of each subinterval.

3. Calculate the area of each rectangle by multiplying its height by its width ($ \Delta x = 1 $).

4. Sum up the areas of all four rectangles to estimate the total area under the graph.

The formula for the area of a rectangle is: $ \text{Area} = \text{height} \times \text{width} $.

So, the estimation of the area under the graph of $ f(x) = 4\sqrt{x} $ from $ x = 0 $ to $ x = 4 $ using four approximating rectangles and right endpoints is the sum of the areas of the four rectangles:

$$
\text{Area} \approx 4\sqrt{1} \times 1 + 4\sqrt{2} \times 1 + 4\sqrt{3} \times 1 + 4\sqrt{4} \times 1
$$