Math / Calculus

QuestionThe following information contains information about a function $f(x)$ at various values of $x$ . \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline $x$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline $f(x)$ & 7 & 2 & 5 & 5 & -1 & 2 & 5 & 3 & 2 \\ \hline \end{tabular}
Use left endpoints and four subintervals to approximate the area beneath the curve of $f(x)$ on the interval $[2,10]$ .
Answer: $\square$ Enter your answer as an exact value (enter as a fraction if necessary).

Studdy Solution

STEP 1

What is this asking? We need to estimate the area under a curve using left endpoints and a specific number of rectangles. Watch out! Make sure to use the left endpoints and the correct width for the rectangles!

STEP 2

1. Find the width of each rectangle.
2. Calculate the area of each rectangle.
3. Sum the areas to find the total approximate area.

STEP 3

We're given the interval $[2, 10]$ and asked to use four subintervals.
To find the width of each rectangle, we subtract the starting $x$ value from the ending $x$ value and divide by the number of subintervals.
This gives us $\frac{10 - 2}{4} = \frac{8}{4} = 2$ .
So, each rectangle has a width of $2$ .

STEP 4

Remember, we're using left endpoints.
The first rectangle starts at $x = 2$ and goes to $x = 2 + 2 = 4$ , so its left endpoint is $x = 2$ .
Its height is $f(2) = 7$ , and its width is $2$ , so its area is $7 \cdot 2 = 14$ .

STEP 5

The second rectangle starts at $x = 4$ and its left endpoint is $x = 4$ .
Its height is $f(4) = 5$ , and its width is $2$ , so its area is $5 \cdot 2 = 10$ .

STEP 6

The third rectangle starts at $x = 6$ and its left endpoint is $x = 6$ .
Its height is $f(6) = -1$ , and its width is $2$ , so its area is $(-1) \cdot 2 = -2$ .

STEP 7

The fourth rectangle starts at $x = 8$ and its left endpoint is $x = 8$ .
Its height is $f(8) = 5$ , and its width is $2$ , so its area is $5 \cdot 2 = 10$ .

STEP 8

Now, we add up the areas of all the rectangles: $14 + 10 + (-2) + 10 = 32$ .

STEP 9

The approximate area beneath the curve is $32$ .

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Studdy Solution

STEP 1

What is this asking? We need to estimate the area under a curve using left endpoints and a specific number of rectangles. Watch out! Make sure to use the left endpoints and the correct width for the rectangles!

STEP 2

1. Find the width of each rectangle.
2. Calculate the area of each rectangle.
3. Sum the areas to find the total approximate area.

STEP 3

STEP 4

Remember, we're using left endpoints.
The first rectangle starts at $x = 2$ and goes to $x = 2 + 2 = 4$ , so its left endpoint is $x = 2$ .
Its height is $f(2) = 7$ , and its width is $2$ , so its area is $7 \cdot 2 = 14$ .

STEP 5

The second rectangle starts at $x = 4$ and its left endpoint is $x = 4$ .
Its height is $f(4) = 5$ , and its width is $2$ , so its area is $5 \cdot 2 = 10$ .

STEP 6

The third rectangle starts at $x = 6$ and its left endpoint is $x = 6$ .
Its height is $f(6) = -1$ , and its width is $2$ , so its area is $(-1) \cdot 2 = -2$ .

STEP 7

The fourth rectangle starts at $x = 8$ and its left endpoint is $x = 8$ .
Its height is $f(8) = 5$ , and its width is $2$ , so its area is $5 \cdot 2 = 10$ .

STEP 8

Now, we add up the areas of all the rectangles: $14 + 10 + (-2) + 10 = 32$ .

STEP 9

The approximate area beneath the curve is $32$ .

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Studdy Solution

STEP 1

What is this asking? We need to estimate the area under a curve using left endpoints and a specific number of rectangles. Watch out! Make sure to use the *left* endpoints and the correct width for the rectangles!

STEP 2

1. Find the width of each rectangle.2. Calculate the area of each rectangle.3. Sum the areas to find the total approximate area.

STEP 3

STEP 4

Remember, we're using *left* endpoints.The **first rectangle** starts at x=2x = 2x=2 and goes to x=2+2=4x = 2 + 2 = 4x=2+2=4, so its left endpoint is x=2x = 2x=2.Its height is f(2)=7f(2) = 7f(2)=7, and its width is 222, so its **area** is 7⋅2=147 \cdot 2 = 147⋅2=14.

STEP 5

The **second rectangle** starts at x=4x = 4x=4 and its left endpoint is x=4x = 4x=4.Its height is f(4)=5f(4) = 5f(4)=5, and its width is 222, so its **area** is 5⋅2=105 \cdot 2 = 105⋅2=10.

STEP 6

The **third rectangle** starts at x=6x = 6x=6 and its left endpoint is x=6x = 6x=6.Its height is f(6)=−1f(6) = -1f(6)=−1, and its width is 222, so its **area** is (−1)⋅2=−2(-1) \cdot 2 = -2(−1)⋅2=−2.

STEP 7

The **fourth rectangle** starts at x=8x = 8x=8 and its left endpoint is x=8x = 8x=8.Its height is f(8)=5f(8) = 5f(8)=5, and its width is 222, so its **area** is 5⋅2=105 \cdot 2 = 105⋅2=10.

STEP 8

Now, we **add up the areas** of all the rectangles: 14+10+(−2)+10=3214 + 10 + (-2) + 10 = 3214+10+(−2)+10=32.

STEP 9

The approximate area beneath the curve is 323232.

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What is this asking? We need to estimate the area under a curve using left endpoints and a specific number of rectangles. Watch out! Make sure to use the left endpoints and the correct width for the rectangles!

1. Find the width of each rectangle.
2. Calculate the area of each rectangle.
3. Sum the areas to find the total approximate area.

Remember, we're using left endpoints.
The first rectangle starts at $x = 2$ and goes to $x = 2 + 2 = 4$ , so its left endpoint is $x = 2$ .
Its height is $f(2) = 7$ , and its width is $2$ , so its area is $7 \cdot 2 = 14$ .

The second rectangle starts at $x = 4$ and its left endpoint is $x = 4$ .
Its height is $f(4) = 5$ , and its width is $2$ , so its area is $5 \cdot 2 = 10$ .

The third rectangle starts at $x = 6$ and its left endpoint is $x = 6$ .
Its height is $f(6) = -1$ , and its width is $2$ , so its area is $(-1) \cdot 2 = -2$ .

The fourth rectangle starts at $x = 8$ and its left endpoint is $x = 8$ .
Its height is $f(8) = 5$ , and its width is $2$ , so its area is $5 \cdot 2 = 10$ .

Now, we add up the areas of all the rectangles: $14 + 10 + (-2) + 10 = 32$ .

The approximate area beneath the curve is $32$ .