Question

Repeat Problem 9 for $f\left(x\right)=4x$

Short answer

(a) The graph of the function $f\left(x\right)=4x$defined on the $[0,6]$interval is

(b) The area of the function by partitioning $[0,6]$into three sub intervals by choosing left end point is $48$.

(c) The area of the function by partitioning $[0,6]$into three sub intervals by choosing right end point is $96$

(d) The area of the function by partitioning $[0,6]$into six sub intervals by choosing left end point is $60$

(e) The area of the function by partitioning $[0,6]$ into six sub intervals by choosing right end point is $84$

(f) The actual area is$72$

Step-by-step solution

Part (a) Step 1. Given

The function:

$f\left(x\right)=4x$

On the interval $[0,6]$

Part (a) Step 2. Graph the function

Graph the given function on the interval $[0,6]$

Part (b) Step 1. Choosing u as left end point

Partition $[0,6]$into three sub intervals, each of length $2$, and choose uas the left endpoint.

The sub intervals are $[0,2],[2,4],[4,6]$

Part (b) Step 2. Approximate the area.

Approximating the area of the function,

$$\begin{gathered} A=f\left(u_1\right)\frac{b-a}{3}+f\left(u_2\right)\frac{b-a}{3}+f\left(u_3\right)\frac{b-a}{3} \\ =f\left(0\right).\left(2\right)+f\left(2\right)\left(2\right)+f\left(4\right)\left(2\right) \\ =0\left(2\right)+8\left(2\right)+16\left(2\right) \\ =48 \end{gathered}$$

Part (c) Step 1. Choosing u as right end point

Partition $[0,6]$into three sub intervals, each of length , and choose uas the right endpoint.

The sub intervals are $[0,2],[2,4],[4,6]$

Part (c) Step 2. Approximate the area.

Approximating the area of the function,

$$\begin{gathered} A=f\left(u_1\right)\frac{b-a}{3}+f\left(u_2\right)\frac{b-a}{3}+f\left(u_3\right)\frac{b-a}{3} \\ =f\left(2\right)\left(2\right)+f\left(4\right)\left(2\right)+f\left(6\right)\left(2\right) \\ =8\left(2\right)+16\left(2\right)+24\left(2\right) \\ =96 \end{gathered}$$

Part (d) Step 1. Choosing u as left end point

Partition $[0,6]$into six sub intervals, each of length $1$, and choose uas the left endpoint.

The sub intervals are $[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]$

Part (d) Step 2. Approximate the area.

Approximating the area of the function,

$A=f\left(u_1\right)\frac{b-a}{6}+f\left(u_2\right)\frac{b-a}{6}+f\left(u_3\right)\frac{b-a}{6}+f\left(u_4\right)\frac{b-a}{6}+f\left(u_5\right)\frac{b-a}{6}+f\left(u_6\right)\frac{b-a}{6}$

$$\begin{gathered} =f\left(0\right)\frac{1}{6}+f\left(1\right)\frac{1}{6}+f\left(2\right)\frac{1}{6}+f\left(3\right)\frac{1}{6}+f\left(4\right)\frac{1}{6}+f\left(5\right)\frac{1}{6} \\ =0+4+8+12+16+20 \\ =60 \end{gathered}$$

Part (e) Step 1. Choosing u as right end point

Partition $[0,6]$into six sub intervals, each of length $1$, and choose uas the right endpoint.

The sub intervals are $[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]$

Part (e) Step 2. Approximate the area.

Approximating the area of the function,

$A=f\left(u_1\right)\frac{b-a}{6}+f\left(u_2\right)\frac{b-a}{6}+f\left(u_3\right)\frac{b-a}{6}+f\left(u_4\right)\frac{b-a}{6}+f\left(u_5\right)\frac{b-a}{6}+f\left(u_6\right)\frac{b-a}{6}$

$$\begin{gathered} =f\left(1\right).1+f\left(2\right).1+f\left(3\right).1+f\left(4\right).1+f\left(5\right).1+f\left(6\right).1 \\ =4.1+8.1+12.1+16.1+20.1+24.1 \\ =84 \end{gathered}$$

Part (f) Step 1. Find the base and height

From the graph, the base is$6$

The height of the triangle is $24$

Part (f) Step 2. Find the area

Area of the triangle $=\frac{1}{2}bh$

$$\begin{gathered} =\frac{1}{2}\times 6\times 24 \\ =72 \end{gathered}$$