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
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Simpsons Rule
`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
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Correct!
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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
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Keep Going!
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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
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Fantastic!
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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
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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
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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
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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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