Math / Calculus

Question田 000162 0003 愺
Question (2) [2 points] a) $[0.8$ points $]$ Approximate the area under the curve for $h(t)$ between $[1,9]$ using a Riemann sum with $n=4$ sub-intervals. Build your approximation so that each rectangle's height is set from the midpoint of its sub-interval. $h(t)=\frac{24}{t}+4 t-12$

Studdy Solution

STEP 1

What is this asking? We need to estimate the area under a curve using rectangles, specifically the midpoint Riemann sum with four rectangles. Watch out! Don't forget to use the midpoint of each sub-interval to determine the rectangle heights, and make sure your calculations are accurate!

STEP 2

1. Find width of each rectangle
2. Find midpoints of each sub-interval
3. Calculate the height of each rectangle
4. Calculate the area of each rectangle
5. Sum the areas

STEP 3

We're given the interval $[1, 9]$ and $n = 4$ rectangles.
To find the width of each rectangle, we subtract the starting point from the endpoint and divide by the number of rectangles.
This gives us $\Delta t = \frac{9 - 1}{4} = \frac{8}{4} = 2$ .
So, each rectangle has a width of 2.

STEP 4

Our sub-intervals are $[1, 3]$ , $[3, 5]$ , $[5, 7]$ , and $[7, 9]$ .
The midpoints are easy to find!
They are 2, 4, 6, and 8, respectively.

STEP 5

We plug each midpoint into our function $h(t) = \frac{24}{t} + 4t - 12$ to get the height of each rectangle.

STEP 6

For $t = 2$ , $h(2) = \frac{24}{2} + 4 \cdot 2 - 12 = 12 + 8 - 12 = 8$ .

STEP 7

For $t = 4$ , $h(4) = \frac{24}{4} + 4 \cdot 4 - 12 = 6 + 16 - 12 = 10$ .

STEP 8

For $t = 6$ , $h(6) = \frac{24}{6} + 4 \cdot 6 - 12 = 4 + 24 - 12 = 16$ .

STEP 9

For $t = 8$ , $h(8) = \frac{24}{8} + 4 \cdot 8 - 12 = 3 + 32 - 12 = 23$ .

STEP 10

Remember, the area of a rectangle is width times height!
Since our width is consistently 2, we multiply each height we just calculated by 2.

STEP 11

Rectangle 1: $2 \cdot 8 = 16$

STEP 12

Rectangle 2: $2 \cdot 10 = 20$

STEP 13

Rectangle 3: $2 \cdot 16 = 32$

STEP 14

Rectangle 4: $2 \cdot 23 = 46$

STEP 15

Finally, we add up the areas of all the rectangles: $16 + 20 + 32 + 46 = 114$ .

STEP 16

The approximate area under the curve $h(t)$ between $[1, 9]$ using the midpoint Riemann sum with four sub-intervals is 114.

Was this helpful?

Studdy Solution

STEP 1

What is this asking? We need to estimate the area under a curve using rectangles, specifically the midpoint Riemann sum with four rectangles. Watch out! Don't forget to use the midpoint of each sub-interval to determine the rectangle heights, and make sure your calculations are accurate!

STEP 2

1. Find width of each rectangle
2. Find midpoints of each sub-interval
3. Calculate the height of each rectangle
4. Calculate the area of each rectangle
5. Sum the areas

STEP 3

We're given the interval $[1, 9]$ and $n = 4$ rectangles.
To find the width of each rectangle, we subtract the starting point from the endpoint and divide by the number of rectangles.
This gives us $\Delta t = \frac{9 - 1}{4} = \frac{8}{4} = 2$ .
So, each rectangle has a width of 2.

STEP 4

Our sub-intervals are $[1, 3]$ , $[3, 5]$ , $[5, 7]$ , and $[7, 9]$ .
The midpoints are easy to find!
They are 2, 4, 6, and 8, respectively.

STEP 5

We plug each midpoint into our function $h(t) = \frac{24}{t} + 4t - 12$ to get the height of each rectangle.

STEP 6

For $t = 2$ , $h(2) = \frac{24}{2} + 4 \cdot 2 - 12 = 12 + 8 - 12 = 8$ .

STEP 7

For $t = 4$ , $h(4) = \frac{24}{4} + 4 \cdot 4 - 12 = 6 + 16 - 12 = 10$ .

STEP 8

For $t = 6$ , $h(6) = \frac{24}{6} + 4 \cdot 6 - 12 = 4 + 24 - 12 = 16$ .

STEP 9

For $t = 8$ , $h(8) = \frac{24}{8} + 4 \cdot 8 - 12 = 3 + 32 - 12 = 23$ .

STEP 10

Remember, the area of a rectangle is width times height!
Since our width is consistently 2, we multiply each height we just calculated by 2.

STEP 11

Rectangle 1: $2 \cdot 8 = 16$

STEP 12

Rectangle 2: $2 \cdot 10 = 20$

STEP 13

Rectangle 3: $2 \cdot 16 = 32$

STEP 14

Rectangle 4: $2 \cdot 23 = 46$

STEP 15

Finally, we add up the areas of all the rectangles: $16 + 20 + 32 + 46 = 114$ .

STEP 16

The approximate area under the curve $h(t)$ between $[1, 9]$ using the midpoint Riemann sum with four sub-intervals is 114.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

What is this asking? We need to estimate the area under a curve using rectangles, specifically the midpoint Riemann sum with four rectangles. Watch out! Don't forget to use the *midpoint* of each sub-interval to determine the rectangle heights, and make sure your calculations are accurate!

STEP 2

1. Find width of each rectangle2. Find midpoints of each sub-interval3. Calculate the height of each rectangle4. Calculate the area of each rectangle5. Sum the areas

STEP 3

STEP 4

Our sub-intervals are [1,3][1, 3][1,3], [3,5][3, 5][3,5], [5,7][5, 7][5,7], and [7,9][7, 9][7,9].The midpoints are easy to find!They are **2**, **4**, **6**, and **8**, respectively.

STEP 5

We plug each midpoint into our function h(t)=24t+4t−12h(t) = \frac{24}{t} + 4t - 12h(t)=t24​+4t−12 to get the height of each rectangle.

STEP 6

For t=2t = 2t=2, h(2)=242+4⋅2−12=12+8−12=8h(2) = \frac{24}{2} + 4 \cdot 2 - 12 = 12 + 8 - 12 = 8h(2)=224​+4⋅2−12=12+8−12=8.

STEP 7

For t=4t = 4t=4, h(4)=244+4⋅4−12=6+16−12=10h(4) = \frac{24}{4} + 4 \cdot 4 - 12 = 6 + 16 - 12 = 10h(4)=424​+4⋅4−12=6+16−12=10.

STEP 8

For t=6t = 6t=6, h(6)=246+4⋅6−12=4+24−12=16h(6) = \frac{24}{6} + 4 \cdot 6 - 12 = 4 + 24 - 12 = 16h(6)=624​+4⋅6−12=4+24−12=16.

STEP 9

For t=8t = 8t=8, h(8)=248+4⋅8−12=3+32−12=23h(8) = \frac{24}{8} + 4 \cdot 8 - 12 = 3 + 32 - 12 = 23h(8)=824​+4⋅8−12=3+32−12=23.

STEP 10

Remember, the area of a rectangle is width times height!Since our width is consistently **2**, we multiply each height we just calculated by **2**.

STEP 11

Rectangle 1: 2⋅8=162 \cdot 8 = 162⋅8=16

STEP 12

Rectangle 2: 2⋅10=202 \cdot 10 = 202⋅10=20

STEP 13

Rectangle 3: 2⋅16=322 \cdot 16 = 322⋅16=32

STEP 14

Rectangle 4: 2⋅23=462 \cdot 23 = 462⋅23=46

STEP 15

Finally, we add up the areas of all the rectangles: 16+20+32+46=11416 + 20 + 32 + 46 = 11416+20+32+46=114.

STEP 16

The approximate area under the curve h(t)h(t)h(t) between [1,9][1, 9][1,9] using the midpoint Riemann sum with four sub-intervals is **114**.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to estimate the area under a curve using rectangles, specifically the midpoint Riemann sum with four rectangles. Watch out! Don't forget to use the midpoint of each sub-interval to determine the rectangle heights, and make sure your calculations are accurate!

1. Find width of each rectangle
2. Find midpoints of each sub-interval
3. Calculate the height of each rectangle
4. Calculate the area of each rectangle
5. Sum the areas

Our sub-intervals are $[1, 3]$ , $[3, 5]$ , $[5, 7]$ , and $[7, 9]$ .
The midpoints are easy to find!
They are 2, 4, 6, and 8, respectively.

We plug each midpoint into our function $h(t) = \frac{24}{t} + 4t - 12$ to get the height of each rectangle.

For $t = 2$ , $h(2) = \frac{24}{2} + 4 \cdot 2 - 12 = 12 + 8 - 12 = 8$ .

For $t = 4$ , $h(4) = \frac{24}{4} + 4 \cdot 4 - 12 = 6 + 16 - 12 = 10$ .

For $t = 6$ , $h(6) = \frac{24}{6} + 4 \cdot 6 - 12 = 4 + 24 - 12 = 16$ .

For $t = 8$ , $h(8) = \frac{24}{8} + 4 \cdot 8 - 12 = 3 + 32 - 12 = 23$ .

Remember, the area of a rectangle is width times height!
Since our width is consistently 2, we multiply each height we just calculated by 2.

Rectangle 1: $2 \cdot 8 = 16$

Rectangle 2: $2 \cdot 10 = 20$

Rectangle 3: $2 \cdot 16 = 32$

Rectangle 4: $2 \cdot 23 = 46$

Finally, we add up the areas of all the rectangles: $16 + 20 + 32 + 46 = 114$ .

The approximate area under the curve $h(t)$ between $[1, 9]$ using the midpoint Riemann sum with four sub-intervals is 114.