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 ( 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 lower - sum

The function \(y = f(x)=4x^{3}\) is increasing on the interval \([0,2]\) (since \(y^\prime=12x^{2}\geq0\) and \(y^\prime = 0\) only at \(x = 0\)). For a lower - sum with \(n = 2\) rectangles, we use the left - hand endpoints.
The left - hand endpoints are \(x_0=0\) and \(x_1 = 1\).
\(f(x_0)=4\times0^{3}=0\) and \(f(x_1)=4\times1^{3}=4\)
The lower - sum \(L=\sum_{i = 0}^{1}f(x_i)\Delta x\)
\(L=(f(0)+f(1))\times\Delta x=(0 + 4)\times1=4\)

Answer:

\(4\)