Use finite approximations to estimate the area under the graph of the function f(x) = 15-x - 2x between x = -5 and x 3 for each of the following cases. a. Using a lower sum with two rectangles of equal width b. Using a lower sum with four rectangles of equal width c. Using an upper sum with two rectangles of equal width d. Using an upper sum with four rectangles of equal width
Use finite approximations to estimate the area under the graph of the function f(x) = 15-x - 2x between x = -5 and x 3 for each of the following cases. a. Using a lower sum with two rectangles of equal width b. Using a lower sum with four rectangles of equal width c. Using an upper sum with two rectangles of equal width d. Using an upper sum with four rectangles of equal width
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I have been quite stuck on this problem could you explain the steps for each part.
![### Estimating Area Under a Curve Using Finite Approximations
To estimate the area under the graph of the function \( f(x) = 15 - x^2 - 2x \) between \( x = -5 \) and \( x = 3 \), we will use finite approximations in four different scenarios. This involves calculating the area using rectangles, either through a lower sum or an upper sum approach, with varying numbers of rectangles.
**a. Lower Sum with Two Rectangles of Equal Width**
In this method, two rectangles are used to approximate the area under the curve. The width of each rectangle is determined by dividing the interval \([-5, 3]\) into two equal parts. The height of each rectangle is determined by the minimum value of the function in each interval.
**b. Lower Sum with Four Rectangles of Equal Width**
Here, we divide the interval \([-5, 3]\) into four equal parts, creating four rectangles. Each rectangle's height is based on the minimum value of the function in its respective interval.
**c. Upper Sum with Two Rectangles of Equal Width**
For the upper sum, two rectangles are used again with the same width as in part (a). However, the height of each rectangle is decided by the maximum value of the function within each interval.
**d. Upper Sum with Four Rectangles of Equal Width**
Finally, the interval \([-5, 3]\) is divided into four equal parts as in part (b). The height of each rectangle is based on the maximum value of the function in its interval.
By using these methods, we can approach the true area under the curve more closely and learn about the utility of different approximation techniques in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a874603-d28a-4420-b369-9729c939fed8%2F52db5cf3-dafa-49b5-9d4e-8237ee984cdc%2Flw9erk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Estimating Area Under a Curve Using Finite Approximations
To estimate the area under the graph of the function \( f(x) = 15 - x^2 - 2x \) between \( x = -5 \) and \( x = 3 \), we will use finite approximations in four different scenarios. This involves calculating the area using rectangles, either through a lower sum or an upper sum approach, with varying numbers of rectangles.
**a. Lower Sum with Two Rectangles of Equal Width**
In this method, two rectangles are used to approximate the area under the curve. The width of each rectangle is determined by dividing the interval \([-5, 3]\) into two equal parts. The height of each rectangle is determined by the minimum value of the function in each interval.
**b. Lower Sum with Four Rectangles of Equal Width**
Here, we divide the interval \([-5, 3]\) into four equal parts, creating four rectangles. Each rectangle's height is based on the minimum value of the function in its respective interval.
**c. Upper Sum with Two Rectangles of Equal Width**
For the upper sum, two rectangles are used again with the same width as in part (a). However, the height of each rectangle is decided by the maximum value of the function within each interval.
**d. Upper Sum with Four Rectangles of Equal Width**
Finally, the interval \([-5, 3]\) is divided into four equal parts as in part (b). The height of each rectangle is based on the maximum value of the function in its interval.
By using these methods, we can approach the true area under the curve more closely and learn about the utility of different approximation techniques in calculus.
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