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 ( square ) 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 \(n = 2\) rectangles, the width \(\Delta x=\frac{2 - 0}{2}=1\).

Step2: Find the right - hand endpoints (for upper sum)

The sub - intervals are \([0,1]\) and \([1,2]\). The right - hand endpoints are \(x_1 = 1\) and \(x_2=2\).

Step3: Calculate the upper sum

The upper sum \(U=\sum_{i = 1}^{2}f(x_i)\Delta x\).
We know that \(f(x)=4x^{3}\), so \(f(1)=4\times1^{3}=4\) and \(f(2)=4\times2^{3}=32\).
Then \(U=(4 + 32)\times1=36\)

Answer:

36