For the following graph of a function, estimate the area under the curve in the interval [-4, 2] using the midpoint approximation and 6 rectangles. Y 4 10 9 8- mm 3 2 1 10 -9 -8 -7 -6 -5 -4 -3 -2 0 1 2 3 4 5 6 Provide your answer below: Area ~ unit2 7 8 9 10
For the following graph of a function, estimate the area under the curve in the interval [-4, 2] using the midpoint approximation and 6 rectangles. Y 4 10 9 8- mm 3 2 1 10 -9 -8 -7 -6 -5 -4 -3 -2 0 1 2 3 4 5 6 Provide your answer below: Area ~ unit2 7 8 9 10
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Estimating Area Under the Curve using Midpoint Approximation**
For the following graph of a function, estimate the area under the curve in the interval \([-4, 2]\) using the midpoint approximation and 6 rectangles.
**Graph Description:**
- The graph has the x-axis ranging from \(-10\) to \(10\) and the y-axis ranging from \(-2\) to \(10\).
- The function appears to be a periodic, sinusoidal wave with multiple peaks and troughs.
**Instructions:**
1. Divide the interval \([-4, 2]\) into 6 equal subintervals.
2. For each subinterval, determine the midpoint.
3. Evaluate the function at each midpoint to determine the height of the rectangle.
4. Calculate the area of each rectangle (height \(\times\) width).
5. Sum the areas to get the total area under the curve.
**Formula:**
The width of each rectangle, \( \Delta x \), is given by:
\[ \Delta x = \frac{(2 - (-4))}{6} = 1 \]
The midpoints for each subinterval are:
- Midpoint 1: \( -3.5 \)
- Midpoint 2: \( -2.5 \)
- Midpoint 3: \( -1.5 \)
- Midpoint 4: \( -0.5 \)
- Midpoint 5: \( 0.5 \)
- Midpoint 6: \( 1.5 \)
**To estimate the area:**
1. Evaluate the function at each midpoint: \( f(-3.5) \), \( f(-2.5) \), \( f(-1.5) \), \( f(-0.5) \), \( f(0.5) \), \( f(1.5) \).
2. Calculate the area for each rectangle as \( f(x_i) \Delta x \).
3. Sum these areas to get the total approximate area under the curve.
**Example Calculation:**
\[ \text{Area} \approx \Delta x [f(-3.5) + f(-2.5) + f(-1.5) + f(-0.5) + f(0.5) + f(1.5)] \]
**Provide your answer below:**
\[ \text{Area} \approx \boxed{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F529e4fe6-e8c4-4722-88aa-75bc8f350b68%2F87567b9b-dfb2-4e8d-8670-e07ced89ea90%2Fd9kwtf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Estimating Area Under the Curve using Midpoint Approximation**
For the following graph of a function, estimate the area under the curve in the interval \([-4, 2]\) using the midpoint approximation and 6 rectangles.
**Graph Description:**
- The graph has the x-axis ranging from \(-10\) to \(10\) and the y-axis ranging from \(-2\) to \(10\).
- The function appears to be a periodic, sinusoidal wave with multiple peaks and troughs.
**Instructions:**
1. Divide the interval \([-4, 2]\) into 6 equal subintervals.
2. For each subinterval, determine the midpoint.
3. Evaluate the function at each midpoint to determine the height of the rectangle.
4. Calculate the area of each rectangle (height \(\times\) width).
5. Sum the areas to get the total area under the curve.
**Formula:**
The width of each rectangle, \( \Delta x \), is given by:
\[ \Delta x = \frac{(2 - (-4))}{6} = 1 \]
The midpoints for each subinterval are:
- Midpoint 1: \( -3.5 \)
- Midpoint 2: \( -2.5 \)
- Midpoint 3: \( -1.5 \)
- Midpoint 4: \( -0.5 \)
- Midpoint 5: \( 0.5 \)
- Midpoint 6: \( 1.5 \)
**To estimate the area:**
1. Evaluate the function at each midpoint: \( f(-3.5) \), \( f(-2.5) \), \( f(-1.5) \), \( f(-0.5) \), \( f(0.5) \), \( f(1.5) \).
2. Calculate the area for each rectangle as \( f(x_i) \Delta x \).
3. Sum these areas to get the total approximate area under the curve.
**Example Calculation:**
\[ \text{Area} \approx \Delta x [f(-3.5) + f(-2.5) + f(-1.5) + f(-0.5) + f(0.5) + f(1.5)] \]
**Provide your answer below:**
\[ \text{Area} \approx \boxed{\
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