Simpsons Rule

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Question 1 of 7

1. Question

Find the area under the curve `y=x^2/4`

`\text(Area) =` (4.6667) `\text(square units)`

Hint

Help Video

Correct

Great Work!

Incorrect

`int_a^b f(x) dx ≈ (b-a)/6 [f(a)+4f((a+b)/2)+f(b)]`

First, construct a table of values for both `x` and `y` given the equation.

`y=x^2/4`

`x`	`2`	`3`	`4`
`y`

Substitute `x=2` into the given equation.

`y`	`=`	$$\frac {\color{#9932CC}{2}^2}{4}$$	Substitute the value of `x`
	`=`	`4/4`	Simplify
`y`	`=`	`1`
`f(a)`	`=`	`1`

`x`	`2`	`3`	`4`
`y`	`1`

Substitute `x=3` into the given equation.

`y`	`=`	$$\frac {\color{#9932CC}{3}^2}{4}$$	Substitute the value of `x`
`y`	`=`	`9/4`	Simplify
`f((a+b)/2)`	`=`	`9/4`

`x`	`2`	`3`	`4`
`y`	`1`	`9/4`

Repeat this process for each `\text(x-value)`

`x`	`2`	`3`	`4`
`y`	`1`	`9/4`	`4`

Apply Simpsons Rule

`A`	`≈`	$$\int_{\color {#9a00c7}{a}}^{\color{#007ddc}{b}} f(x) dx$$

	`≈`	$$ \frac{\color{#007ddc}{b}- \color{#9a00c7}{a}}{6} [\color {#00880a}{f(a)} +4 \color{#00880a}{\frac{f(a+b)}{2}}+\color {#00880a}{f(b)}]$$	Simpsons Rule formula

	`≈`	$$ \frac{\color{#007ddc}{4}- \color{#9a00c7}{2}}{6} [\color {#00880a}{1} +4 \color{#00880a}{(\frac{9}{4})}+\color {#00880a}{4}]$$	Substitute known values

	`≈`	`2/6 [5+36/4]`	Simplify

	`≈`	`1/3 [14]`

	`≈`	`14/3`

	`≈`	`4.6667`

`4.6667 \text(square units)`

Question 2 of 7

2. Question

Find the area under the curve `y=2^x`

`\text(Area) =` (43.333)

Hint

Help Video

Correct

Correct!

Incorrect

Simpsons Rule

`int_a^b f(x) dx ≈ h/3 [y_0+y_L+2(y_2+y_4+…)+4(y_1+y_3+…)]`

First, find the values of `a`, `b` and `n` from the given equation

$$\int_{\color{#9a00c7}{1}}^{\color{#007ddc}{5}} y$$

`=`

`2^x`

`a=1`(lower limit)

`b=5`(upper limit)

`n=4`(number of strips in given diagram)

Solve for `h`

`h`	`=`	$$\frac {\color {#007ddc}{b}-\color {#9a00c7}{a}}{\color {#00880a}{n}} $$

	`=`	$$\frac {\color {#007ddc}{5}-\color {#9a00c7}{1}}{\color {#00880a}{4}} $$	Substitute values of `a`, `b`, and `n`

	`=`	`4/4`	Simplify

`h`	`=`	`1`

First, construct a table of values for both `x` and `y` given the equation.

`y=2^x`

`x`	`1`	`2`	`3`	`4`	`5`
`y`

Substitute `x=1` into the given equation.

`y`	`=`	$$ 2^{\color{#9932CC}{1}}$$	Substitute the value of `x`
	`=`	`2^1`	Simplify
`y_0`	`=`	`2`

`x`	`1`	`2`	`3`	`4`	`5`
`y`	`2`

Substitute `x=2` into the given equation.

`y`	`=`	$$2^{\color{#9932CC}{2}}$$	Substitute the value of `x`
`y_1`	`=`	`4`	Simplify

`x`	`1`	`2`	`3`	`4`	`5`
`y`	`2`	`4`

Repeat this process for each `\text(x-value)`

`x`	`1`	`2`	`3`	`4`	`5`
`y`	`2`	`4`	`8`	`16`	`32`

Apply Simpsons Rule

`A`	`≈`	$$\int_{\color {#9a00c7}{a}}^{\color{#007ddc}{b}} f(x) dx$$

	`≈`	`h/3 [``y_0``+``y_L``+2(``y_2``+``y_4``+…)+4(``y_1``+``y_3``+…)]`	Simpsons Rule formula

	`≈`	`1/3 [``2``+``32``+2(``8``)+4(``4``+``16``)]`	Substitute known values

	`≈`	`1/3 [34+16+80]`	Simplify

	`≈`	`1/3 [130]`

	`≈`	`(130)/3`

	`≈`	`43.333`

`43.333`

Question 3 of 7

3. Question

Find the area under the curve `y=6/x`

`\text(Area) =` (4.159)

Hint

Help Video

Correct

Keep Going!

Incorrect

Simpsons Rule

`int_a^b f(x) dx ≈ h/3 [y_0+y_L+2(y_2+y_4+…)+4(y_1+y_3+…)]`

First, find the values of `a`, `b` and `n` from the given equation

$$\int_{\color{#9a00c7}{3}}^{\color{#007ddc}{6}} y$$

`=`

`6/x`

`a=3`(lower limit)

`b=6`(upper limit)

`n=6`(number of strips in given diagram)

Solve for `h`

`h`	`=`	$$\frac {\color {#007ddc}{b}-\color {#9a00c7}{a}}{\color {#00880a}{n}} $$

	`=`	$$\frac {\color {#007ddc}{6}-\color {#9a00c7}{3}}{\color {#00880a}{6}} $$	Substitute values of `a`, `b`, and `n`

	`=`	`3/6`	Simplify

`h`	`=`	`1/2`

First, construct a table of values for both `x` and `y` given the equation.

`y=2^x`

`x`	`3`	`3.5`	`4`	`4.5`	`5`	`5.5`	`6`
`y`

Substitute `x=3` into the given equation.

`y`	`=`	$$\frac {6}{\color{#9932CC}{3}}$$	Substitute the value of `x`
	`=`	`2`	Simplify
`y_0`	`=`	`2`

`x`	`3`	`3.5`	`4`	`4.5`	`5`	`5.5`	`6`
`y`	`2`

Substitute `x=3.5` into the given equation.

`y`	`=`	$$\frac{6}{\color{#9932CC}{3.5}}$$	Substitute the value of `x`
`y_1`	`=`	`6/3.5`	Simplify

`x`	`3`	`3.5`	`4`	`4.5`	`5`	`5.5`	`6`
`y`	`2`	`6/3.5`

Repeat this process for each `\text(x-value)`

`x`	`3`	`3.5`	`4`	`4.5`	`5`	`5.5`	`6`
`y`	`2`	`6/3.5`	`6/4`	`6/4.5`	`6/5`	`6/5.5`	`1`

Apply Simpsons Rule

`A`	`≈`	$$\int_{\color {#9a00c7}{a}}^{\color{#007ddc}{b}} f(x) dx$$

	`≈`	`h/3 [``y_0``+``y_L``+2(``y_2``+``y_4``+…)+4(``y_1``+``y_3``+…)]`	Simpsons Rule formula

	`≈`	`(1/2)/3 [``2``+``1``+2(``6/4``+``6/5``)+4(``6/3.5``+``6/4.5``+``6/5.5``)]`	Substitute known values

	`≈`	`1/6 [3+2(27/10)+4(4.1385)]`	Simplify

	`≈`	`1/6 [3+54/10+16.554113]`

	`≈`	`1/6 (24.954113)`

	`≈`	`4.159`

`4.159`

Question 4 of 7

4. Question

Find and estimate the area of the orange parcel of land.

`\text(Area) =` (1960) `m^2`

Hint

Help Video

Correct

Fantastic!

Incorrect

Simpsons Rule

`int_a^b f(x) dx ≈ h/3 [y_0+y_L+2(y_2+y_4+…)+4(y_1+y_3+…)]`

First, find the value of `h` from the given illustration.

`h`

`=`

`20` `m`

Next, identify the values for substitution into Simpsons Rule.

`y_0`	`y_1`	`y_2`	`y_3`	`y_L`
`21.5`	`24`	`22.5`	`28`	`19.5`

Apply Simpsons Rule

`A`	`≈`	`h/3 [``y_0``+``y_L``+2(``y_2``+``y_4``+…)+4(``y_1``+``y_3``+…)]`	Simpsons Rule formula

	`≈`	`(20)/3 [``21.5``+``19.5``+2(``22.5``)+4(``24``+``28``)]`	Substitute known values

	`≈`	`20/3 [41+45+208]`	Simplify

	`≈`	`20/3 [294]`

	`≈`	`1960`

`1960` `m^2`

Question 5 of 7

5. Question

Find and estimate the area of the huge block of land.

`\text(Area) =` (251333 1/3, 251333.33, 251333.3333) `m^2`

Hint

Help Video

Correct

Excellent!

Incorrect

Simpsons Rule

`int_a^b f(x) dx ≈ h/3 [y_0+y_L+2(y_2+y_4+…)+4(y_1+y_3+…)]`

First, find the value of `h` from the given illustration.

`h`

`=`

`200` `m`

Next, identify the values for substitution into Simpsons Rule.

`y_0`	`y_1`	`y_2`	`y_3`	`y_4`	`y_5`	`y_L`
`0`	`150`	`290`	`320`	`210`	`180`	`170`

Apply Simpsons Rule

`A`	`≈`	`h/3 [``y_0``+``y_L``+2(``y_2``+``y_4``+…)+4(``y_1``+``y_3``+…)]`	Simpsons Rule formula

	`≈`	`(200)/3 [``0``+``170``+2(``290``+``210``)+4(``150``+``320``+``180``)]`	Substitute known values

	`≈`	`200/3 [170+1000+4(2600)]`	Simplify

	`≈`	`200/3 [3770]`

	`≈`	`251333 1/3`

`251333 1/3` `m^2`

Question 6 of 7

6. Question

Find and estimate the area of the yellow parcel of land.

`\text(Area) =` (45000) `m^2`

Hint

Help Video

Correct

Nice Job!

Incorrect

Simpsons Rule

`int_a^b f(x) dx ≈ h/3 [y_0+y_L+2(y_2+y_4+…)+4(y_1+y_3+…)]`

First, find the value of `h` from the given illustration.

`h`	`=`	`400/4`

`h`	`=`	`100` `m`

Next, identify the values for substitution into Simpsons Rule.

`y_0`	`y_1`	`y_2`	`y_3`	`y_L`
`110`	`98`	`102`	`124`	`148`

Apply Simpsons Rule

`A`	`≈`	`h/3 [``y_0``+``y_L``+2(``y_2``+``y_4``+…)+4(``y_1``+``y_3``+…)]`	Simpsons Rule formula

	`≈`	`(100)/3 [``110``+``148``+2(``102``)+4(``98``+``124``)]`	Substitute known values

	`≈`	`100/3 [258+204+888]`	Simplify

	`≈`	`100/3 [1350]`

	`≈`	`45000`

`45000` `m^2`

Question 7 of 7

7. Question

Find and estimate the area of the irregular piece of land.

`\text(Area) =` (5558 2/3) `m^2`

Hint

Help Video

Correct

Well Done!

Incorrect

Simpsons Rule

`int_a^b f(x) dx ≈ h/3 [d_f+4d_m+d_L]`

First, find the value of `h` from the given illustration.

`h`

`=`

`22` `m`

Next, identify the values for substitution into Simpsons Rule.

`d_f`	`d_m`	`d_L`
`0`	`49`	`68`

`d_f`	`d_m`	`d_L`
`68`	`88`	`74`

Apply Simpsons Rule to find `A_1`

`A_1`	`≈`	`h/3 [d_f+4d_m+d_L]`	Simpsons Rule formula

	`≈`	`(22)/3 [``0``+4(``49``)+``68``]`	Substitute known values

	`≈`	`22/3 [68+196]`	Simplify

	`≈`	`22/3 [264]`

`A_1`	`≈`	`1936` `m^2`

Apply Simpsons Rule to find `A_2`

`A_2`	`≈`	`h/3 [d_f+4d_m+d_L]`	Simpsons Rule formula

	`≈`	`(22)/3 [``68``+4(``88``)+``74``]`	Substitute known values

	`≈`	`22/3 [142+352]`	Simplify

	`≈`	`22/3 [494]`

`A_2`	`≈`	`3622 2/3` `m^2`

Add `A_1` and `A_2`

`A`	`≈`	`A_1``+``A_2`	Combine the areas

	`≈`	`1936``+``3622 2/3`

	`≈`	`5558 2/3`

`5558 2/3` `m^2`

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