If #f(x)=x^(1/2)#, #1 <= x <= 4# approximate the area under the curve using ten approximating rectangles of equal widths and left endpoints?
Area
We have:
# f(x) = sqrt(x) #
We want to calculate over the interval
# Deltax = (41)/10 = 0.3#
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_("left") f(x)Deltax #
# " " = (0.3) { f(1)+f(1.3) + f(1.6) + ... +f(3.4) + f(3.7) } \ \ \ # (The LHS values)
# " " = 0.3*(1+1.140175425+1.264911064+1.378404875+#
# " " 1.483239697+1.58113883+1.673320053+#
# " " 1.760681686+1.843908891+1.923538406#
# " " = 0.3*15.04931893 #
# " " = 4.514795679 #
Actual Value
For comparison of accuracy:
# Area = int_1^4 \ x^(1/2) \ dx #
# " " = [(x^(3/2))/(3/2)]_1^4 #
# " " = 2/3[x^(3/2)]_1^4 #
# " " = 2/3{(8)  (1)} #
# " " = 14/3 #
# " " ~~ 4.6667 #
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To approximate the area under the curve of ( f(x) = \sqrt{x} ) from ( x = 1 ) to ( x = 4 ) using ten approximating rectangles with left endpoints, follow these steps:

Calculate the width of each rectangle: ( \Delta x = \frac{b  a}{n} ), where ( a = 1 ) (lower limit), ( b = 4 ) (upper limit), and ( n = 10 ) (number of rectangles).

Compute the height of each rectangle using the left endpoint of each subinterval: ( f(a + i\Delta x) ), where ( i = 0, 1, 2, ..., n1 ).

Multiply the width and height of each rectangle to find the area of each rectangle.

Sum up the areas of all rectangles to get the approximation of the area under the curve.
So, the steps are:

( \Delta x = \frac{4  1}{10} = \frac{3}{10} ).

( f(x) = \sqrt{x} ). ( f(a + i\Delta x) = \sqrt{1 + i\left(\frac{3}{10}\right)} ), for ( i = 0, 1, 2, ..., 9 ).

Calculate the area of each rectangle: ( \text{Area}_i = \Delta x \times f(a + i\Delta x) ), for ( i = 0, 1, 2, ..., 9 ).

Sum up all the areas: ( \text{Approximated Area} = \sum_{i=0}^{9} \text{Area}_i ).
After computing these steps, you'll get the approximation of the area under the curve using ten approximating rectangles of equal widths and left endpoints.
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