**Estimating the Area Under a Curve**To estimate the area under the curve of the function $ f(x) = x^3 - 4x $ on the interval $[-1, 3]$, we will calculate the Left Riemann Sum ($L_8$) and the Right Riemann Sum ($R_8$).**1. Understand the Function:**The function given is $ f(x) = x^3 - 4x $. This cubic function has both positive and negative values within the interval $[-1, 3]$.**2. Interval Breakdown:**The interval $[-1, 3]$ is split into 8 equal subintervals (as denoted by the subscript 8 in $L_8$ and $R_8$). Each subinterval width ($\Delta x$) can be calculated as:\[ \Delta x = \frac{(3 - (-1))}{8} = \frac{4}{8} = 0.5 \]**3. Calculate $L_8$ (Left Riemann Sum):**For $L_8$, we use the left endpoints of each subinterval to estimate the area. The left endpoints for the 8 subintervals from $[-1, 3]$ are:\[ -1, -0.5, 0, 0.5, 1, 1.5, 2, 2.5 \]The left Riemann sum is then calculated by:\[ L_8 = \sum_{i=0}^{7} f(x_i) \Delta x \]where $ x_i = -1 + i \Delta x $.**4. Calculate $R_8$ (Right Riemann Sum):**For $R_8$, we use the right endpoints of each subinterval to estimate the area. The right endpoints for the 8 subintervals from $[-1, 3]$ are:\[ -0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3 \]The right Riemann sum is then calculated by:\[ R_8 = \sum_{i=1}^{8} f(x_i) \Delta x \]where \( x_i = -1 + i \Delta x

To find area under a curve y=f(x). Integrate f(x) from x = a to x = b.

Answered: Estimate the area under the curve of f…

Math

Calculus

Estimate the area under the curve of f (x) = x3 – 4x on [-1, 3] by calculating Lg and Rg.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter7: Integration

Section7.2: Substitution

Problem 1E: Integration by substitution is related to what differentiation method? What type of integrand...

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Question

**Estimating the Area Under a Curve**

To estimate the area under the curve of the function $ f(x) = x^3 - 4x $ on the interval $[-1, 3]$, we will calculate the Left Riemann Sum ($L_8$) and the Right Riemann Sum ($R_8$).

**1. Understand the Function:**
The function given is $ f(x) = x^3 - 4x $. This cubic function has both positive and negative values within the interval $[-1, 3]$.

**2. Interval Breakdown:**
The interval $[-1, 3]$ is split into 8 equal subintervals (as denoted by the subscript 8 in $L_8$ and $R_8$). Each subinterval width ($\Delta x$) can be calculated as:
\[ \Delta x = \frac{(3 - (-1))}{8} = \frac{4}{8} = 0.5 \]

**3. Calculate $L_8$ (Left Riemann Sum):**
For $L_8$, we use the left endpoints of each subinterval to estimate the area. The left endpoints for the 8 subintervals from $[-1, 3]$ are:
\[ -1, -0.5, 0, 0.5, 1, 1.5, 2, 2.5 \]

The left Riemann sum is then calculated by:
\[ L_8 = \sum_{i=0}^{7} f(x_i) \Delta x \]
where $ x_i = -1 + i \Delta x $.

**4. Calculate $R_8$ (Right Riemann Sum):**
For $R_8$, we use the right endpoints of each subinterval to estimate the area. The right endpoints for the 8 subintervals from $[-1, 3]$ are:
\[ -0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3 \]

The right Riemann sum is then calculated by:
\[ R_8 = \sum_{i=1}^{8} f(x_i) \Delta x \]
where \( x_i = -1 + i \Delta x

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