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 9 square units.
(type an integer or a decimal. simplify your answer.)

c. the estimated area using an upper sum with two rectangles of equal width is 36 square units
(type an integer or a decimal. simplify your answer.)

d. the estimated area using an upper 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

The interval is from \(x = 0\) to \(x=2\). For \(n = 4\) rectangles, the width \(\Delta x=\frac{2 - 0}{4}=0.5\)

Step2: Determine the sub - intervals

The sub - intervals are \([0,0.5]\), \([0.5,1]\), \([1,1.5]\), \([1.5,2]\)

Step3: Find the upper sum

For an upper sum, we use the right - hand endpoints of each sub - interval.
The right - hand endpoints are \(x_1 = 0.5\), \(x_2=1\), \(x_3 = 1.5\), \(x_4=2\)

Step4: Calculate the function values at the right - hand endpoints

\(f(x_1)=4\times(0.5)^3=4\times0.125 = 0.5\)
\(f(x_2)=4\times(1)^3=4\)
\(f(x_3)=4\times(1.5)^3=4\times3.375 = 13.5\)
\(f(x_4)=4\times(2)^3=32\)

Step5: Calculate the upper sum

\(S=\sum_{i = 1}^{4}f(x_i)\Delta x\)
\(S=(0.5 + 4+13.5 + 32)\times0.5\)
\(S=(50)\times0.5\)
\(S = 25\)

Answer:

\(25\)