Q. 4 - Calculus

Question

Consider the area between the graph of a function $f$and the $X$-axis from $x=0$ to $x=2$, as shown in each of the two figures that follow.

(a) Use the grid on the left and whatever method you like to approximate this area. Then use the grid on the right and the same method to make another approximation. Which approximation is likely more accurate, and why?

(b) Use the grid at the left to get an upper bound on the area of the region. In other words, make an approximation that you know is greater than the actual area. Repeat for the grid at the right.

(c) Use the grid at the left to get a lower bound on the area of the region. In other words, make an approximation that you know is less than the actual area. Repeat for the grid at the right.

(d) Use your answers to parts (a)–(c) to come up with your best possible guess for the actual area under the curve.

Step-by-Step Solution

Answer

1. The area of the left figure is roughly $1.96875$, and the area of the right figure is roughly $1.984375$.

   Because the square grids on the right are finer than the left, the area of the grid on the right has a more accurate approximation than the grid on the right.

2. The area upper bounds for the left graph and the right graph are $2.0625$and $2.015625$, respectively.

3. The area lower bounds for the left graph and the right graph are $1.84375$ and $1.8984$, respectively.

4. The actual area is two square units.

Part (a) Step 1: Given information

The region from $x=0$ to $x=2$ between the graph of a function $f$ and the $x$-axis, as depicted in each of the following two pictures.

Part (a) Step 2: Calculation

Consider the following graph is,

By counting the square grids, we will determine the area.

The grid is divided into $\frac{1}{4}$ units on each side.

Therefore, the area of each square grid is ${\left(\frac{1}{4}\right)}^2=\frac{1}{16}$ square units.

Currently, there are $26$ complete squares in the shaded region of the depicted figure.

Area of 26 completed square is $26\times \frac{1}{16}=1.625$.

There are three squares that are just partially filled.

The Area of three half-filled square is $\left(3\times \frac{1}{16}\right)\times \frac{1}{2}=0.09375$.

There are $4$ squares that are more than half-filled.

Four half-filled squares' surface area is$4\times \frac{1}{16}=0.25$.

Ignore the square that is only partially completed.

Total area $=1.625+0.09375+0.25$

$=1.96875$

Consider the above graph (right graph), we will find the area by the method of counting the square grids.

Each side of the grid is measured as $\frac{1}{8}$ units.

Therefore, the area of each square grid is ${\left(\frac{1}{8}\right)}^2=\frac{1}{64}$ square units.

Now, the number of complete square in the shaded area of the given figure is $116$

Area of 116 completed square is $116\times \frac{1}{64}=1.8125$.

The number of half-filled square is 2 .

Area of 2 half-filled square is $\left(2\times \frac{1}{64}\right)\times \frac{1}{2}=0.03125$.

The number of more than half-filled square is 9 .

Area of 9 half-filled square is $9\times \frac{1}{64}=0.140625$.

Ignore the square less than half-filled.

Total area $=1.8125+0.03125+0.140625$

$=1.984375$

The area of the grid on the right has more accurate approximation that the grid on the right because the square grids on the right are more fine than the left.

Hence, the approximate area of the left figure is $1.96875$ and the right figure is $1.984375$ .

The area of the grid on the right has more accurate approximation that the grid on the right because the square grids on the right are more fine than the left.

Part (b) Step 1: Given information

The area between the graph of a function $f$and the $x$-axis from $x=0$ to $x=2$, as shown in each of the two figures that follow.

Part (b) Step 2: Calculation

Consider the following graph ,

Each square grid's area is ${\left(\frac{1}{4}\right)}^2=\frac{1}{16}$ square units.

Now, the number of squares included in the shaded area of the accompanying figure, excluding those that are less than half-filled, is $33$

Area of $33$those square is $33\times \frac{1}{16}=2.0625$.

Consequently, the area for the left graph's upper bound is $2.0625$.

Take a look at the graph on the right, where the area of each square grid is  square units.

Now, there are $129$ squares in the shaded area of the above figure, excluding the squares that are less than half-filled.

Area of $129$ those square is $129·\frac{1}{64}=2.015625$.

Thus, the upper bound on the area for the right graph is \(2.015625\).

Hence, the upper bound on the area for the left graph is \(2.0625\) and for the right graph is$2.015625$.

Part (c) Step 1: Given information

The area between the graph of a function $f$and the $x$-axis from $x=0$ to $x=2$, as shown in each of the two figures that follow.

Part (c) Step 2: Calculation

Consider the graph,

The each square grid area is ${\left(\frac{1}{4}\right)}^2=\frac{1}{16}$ square units.

The darkened region of the given figure now has $26$ worth of complete squares.

Area of those 26 square is$26\times \frac{1}{16}=1.625$.

Half-filled and more than half-filled squares are represented by a certain number $7$

Area of those 7 square is $\left(7\times \frac{1}{16}\right)\times \frac{1}{2}=0.21875$.

The area's lower bound is$1.625+0.21875=1.84375$.

Consequently, the area for the left graph's lower bound is$1.84375$.

Consider the graph on the right, where each square grid's area is ${\left(\frac{1}{8}\right)}^2=\frac{1}{64}$square units.

Now, there are $116$ full squares in the darkened region in the given figure.

The area of those 116 squares is $116\times \frac{1}{64}=1.8125$

There are$11$ squares with more than half of them filled.

Area of those 11 square is $\left(11\times \frac{1}{64}\right)\times \frac{1}{2}=0.0859375$.

Lower bound on the area is $1.84375+0.08594=1.8984$

Therefore,$1.8984$ is the lower bound on the area for the left graph.

Consequently, the area's lower bounds for the left graph and the right graph are $1.84375$ and $1.8984$, respectively.

Part (d) Step 1: Given information

The region from $x=0$ to $x=2$ between the graph of a function $f$ and the $x$-axis, as depicted in each of the following two pictures.

Part (d) Step 2: Calculation

The area of the left figure is roughly $1.96875$, and the area of the right figure is roughly $1.984375$.

The area upper bounds for the left graph and the right graph are $2.0625$ and $2.015625$, respectively.

The area's bottom bounds for the left graph and the right graph are $1.84375$ and$1.8984$, respectively.

The actual area must thus be two square units.

It follows that the curve's real area under the curve is two square units.