Question

estimate the area under the graph of ( f(x) = 4x^{3} ) between ( x = 0 ) and ( x = 2 ) using each finite approximation below

a. a lower sum with two rectangles of equal width
b. a lower sum with four rectangles of equal width
c. an upper sum with two rectangles of equal width
d. an upper sum with four rectangles of equal width

a. the estimated area using a lower sum with two rectangles of equal width is 4 square units
(type an integer or a decimal. simplify your answer.)

b. the estimated area using a lower sum with four rectangles of equal width is square units
(type an integer or a decimal. simplify your answer.)

Explanation:

Step1: Calculate the width of each rectangle

For \(n = 4\) rectangles, the width \(\Delta x=\frac{2 - 0}{4}=0.5\). The sub - intervals are \([0,0.5]\), \([0.5,1]\), \([1,1.5]\), \([1.5,2]\).
Since \(y = f(x)=4x^{3}\) is an increasing function on the interval \([0,2]\) (because \(y^\prime=12x^{2}\geq0\) and \(y^\prime = 0\) only at \(x = 0\)), for a lower sum, we use the left - hand endpoints of each sub - interval.

Step2: Calculate the function values at the left - hand endpoints

The left - hand endpoints are \(x_0 = 0\), \(x_1=0.5\), \(x_2 = 1\), \(x_3=1.5\).
\(f(x_0)=4\times0^{3}=0\)
\(f(x_1)=4\times(0.5)^{3}=4\times\frac{1}{8}=0.5\)
\(f(x_2)=4\times1^{3}=4\)
\(f(x_3)=4\times(1.5)^{3}=4\times3.375 = 13.5\)

Step3: Calculate the lower sum

The lower sum \(L=\sum_{i = 0}^{3}f(x_i)\Delta x\)
\(L=(0 + 0.5+4 + 13.5)\times0.5\)
\(L=(18)\times0.5=9\)

Answer:

\(9\)