Let's go step-by-step through the process for the function \( f(x) = -x^2 + 6x - 7 \) over the interval \([2, 4]\): --- ### **Step 1: Sketch the graph and shade the area** - The function \( f(x) = -x^2 + 6x - 7 \) is a downward-opening parabola. - You're interested in the area under the curve from \( x = 2 \) to \( x = 4 \) and above the x-axis. --- ### **Step 2: Write the definite integral** The area under the curve from \( x=2 \) to \( x=4 \) is: \[ \int_{2}^{4} (-x^2 + 6x - 7) \, dx \] --- ### **Step 3: Left endpoint rectangle approximation with \( n = 4 \)** - Interval: \([2, 4]\) - Width: \( \Delta x = \frac{4-2}{4} = 0.5 \) - Left endpoints: \( x_0 = 2, x_1 = 2.5, x_2 = 3, x_3 = 3.5 \) Evaluate \( f(x) \) at these points: \[ \begin{align*} f(2) &= -(2)^2 + 6(2) - 7 = -4 + 12 - 7 = 1 \\ f(2.5) &= -(2.5)^2 + 6(2.5) - 7 = -6.25 + 15 - 7 = 1.75 \\ f(3) &= -(3)^2 + 6(3) - 7 = -9 + 18 - 7 = 2 \\ f(3.5) &= -(3.5)^2 + 6(3.5) - 7 = -12.25 + 21 - 7 = 1.75 \\ \end{align*} \] Table: | i | \( x_i \) | \( f(x_i) \) | |---|-----------|--------------| | 0 | 2.0 | 1.00 | | 1 | 2.5 | 1.75 | | 2 | 3.0 | 2.00 | | 3 | 3.5 | 1.75 | --- ### **Step 4: Compute the area estimate (\( n = 4 \), left endpoint)** \[ \text{Area} \approx \Delta x \left[ f(2) + f(2.5) + f(3) + f(3.5) \right] = 0.5 \times (1 + 1.75 + 2 + 1.75) = 0.5 \times 6.5 = 3.25 \] --- ### **Step 5: Left endpoint rectangle approximation with \( n = 8 \)** - Width: \( \Delta x = \frac{4-2}{8} = 0.25 \) - Left endpoints: \( x_0 = 2, x_1 = 2.25, x_2 = 2.5, x_3 = 2.75, x_4 = 3, x_5 = 3.25, x_6 = 3.5, x_7 = 3.75 \) Calculate \( f(x) \) at each: \[ \begin{align*} f(2) &= 1 \\ f(2.25) &= -(2.25)^2 + 6(2.25) - 7 = -5.0625 + 13.5 - 7 = 1.4375 \\ f(2.5) &= 1.75 \\ f(2.75) &= -(2.75)^2 + 6(2.75) - 7 = -7.5625 + 16.5 - 7 = 1.9375 \\ f(3) &= 2 \\ f(3.25) &= -(3.25)^2 + 6(3.25) - 7 = -10.5625 + 19.5 - 7 = 1.9375 \\ f(3.5) &= 1.75 \\ f(3.75) &= -(3.75)^2 + 6(3.75) - 7 = -14.0625 + 22.5 - 7 = 1.4375 \\ \end{align*} \] Table: | i | \( x_i \) | \( f(x_i) \) | |---|-----------|--------------| | 0 | 2.00 | 1.00 | | 1 | 2.25 | 1.44 | | 2 | 2.50 | 1.75 | | 3 | 2.75 | 1.94 | | 4 | 3.00 | 2.00 | | 5 | 3.25 | 1.94 | | 6 | 3.50 | 1.75 | | 7 | 3.75 | 1.44 | --- ### **Step 6: Area estimate for \( n = 8 \)** Add up the values: \[ \text{Sum} = 1 + 1.4375 + 1.75 + 1.9375 + 2 + 1.9375 + 1.75 + 1.4375 = 13.25 \] \[ \text{Area} \approx 0.25 \times 13.25 = 3.3125 \] --- ### **Step 7: Vocabulary** - **Definite Integral**: The exact value of the area under the curve between two points. - **Approximate Integral**: Using methods like rectangles (Riemann sums) to estimate the area. - **Antiderivative**: A function whose derivative is the given function (used to compute definite integrals). --- ### **Step 8: Find the actual area using the Fundamental Theorem of Calculus** First, find the antiderivative \( F(x) \): \[ \int (-x^2 + 6x - 7) dx = -\frac{1}{3}x^3 + 3x^2 - 7x + C \] Evaluate from 2 to 4: \[ F(4) = -\frac{1}{3}(4)^3 + 3(4)^2 - 7(4) = -\frac{64}{3} + 48 - 28 = -\frac{64}{3} + 20 \] \[ F(2) = -\frac{1}{3}(8) + 3(4) - 7(2) = -\frac{8}{3} + 12 - 14 = -\frac{8}{3} - 2 \] Now, subtract: \[ \text{Area} = F(4) - F(2) = \left(-\frac{64}{3} + 20\right) - \left(-\frac{8}{3} - 2\right) = -\frac{64}{3} + 20 + \frac{8}{3} + 2 = \left(-\frac{64}{3} + \frac{8}{3}\right) + (20 + 2) = -\frac{56}{3} + 22 \] \[ 22 - \frac{56}{3} = \frac{66}{3} - \frac{56}{3} = \frac{10}{3} \approx 3.333 \] --- ## **Final Answers** 1. **Definite integral for area:** \( \int_2^4 (-x^2 + 6x - 7)\,dx \) 2. **Left endpoint estimate, \( n=4 \):** \( 3.25 \) 3. **Left endpoint estimate, \( n=8 \):** \( 3.3125 \) 4. **Actual area:** \( \frac{10}{3} \approx 3.333 \) Let me know if you'd like a graph, more explanation, or calculations for right endpoints!

Certainly! Let's walk through the entire problem step-by-step with detailed calculations, explanations, and the final answers. --- ### **Given function:** \[ f(x) = -x^2 + 6x - 7 \] **Interval:** \([2, 4]\) --- ### **Step 1: Sketch and understand the graph** - The function is a downward-opening parabola because the coefficient of \(x^2\) is negative. - Our goal is to find the area under this curve from \(x=2\) to \(x=4\). --- ### **Step 2: Write the definite integral** The exact area under \(f(x)\) from 2 to 4 is: \[ \boxed{ \int_{2}^{4} (-x^2 + 6x - 7) \, dx } \] --- ### **Step 3: Left endpoint rectangle approximation with \( n=4 \)** - **Number of rectangles:** \( n = 4 \) - **Width of each rectangle:** \[ \Delta x = \frac{b - a}{n} = \frac{4 - 2}{4} = 0.5 \] - **Left endpoints:** \[ x_0 = 2,\quad x_1 = 2.5,\quad x_2 = 3,\quad x_3 = 3.5 \] - **Evaluate \(f(x)\) at these points:** 1. \(f(2)\): \[ f(2) = - (2)^2 + 6 \times 2 - 7 = -4 + 12 - 7 = 1 \] 2. \(f(2.5)\): \[ f(2.5) = - (2.5)^2 + 6 \times 2.5 - 7 = -6.25 + 15 - 7 = 1.75 \] 3. \(f(3)\): \[ f(3) = -9 + 18 - 7 = 2 \] 4. \(f(3.5)\): \[ f(3.5) = -12.25 + 21 - 7 = 1.75 \] - **Table of values:** | \(i\) | \(x_i\) | \(f(x_i)\) | |--------|---------|------------| | 0 | 2.0 | 1.00 | | 1 | 2.5 | 1.75 | | 2 | 3.0 | 2.00 | | 3 | 3.5 | 1.75 | - **Approximate area using left endpoints:** \[ \text{Area} \approx \Delta x \times \left( f(x_0) + f(x_1) + f(x_2) + f(x_3) \right) \] \[ = 0.5 \times (1 + 1.75 + 2 + 1.75) = 0.5 \times 6.5 = \boxed{3.25} \] --- ### **Step 4: Left endpoint rectangle approximation with \( n=8 \)** - **Number of rectangles:** \( n=8 \) - **Width:** \[ \Delta x = \frac{4 - 2}{8} = 0.25 \] - **Left endpoints:** \[ x_i = 2 + i \times 0.25,\quad i=0,1,\dots,7 \] \[ x_0=2,\quad x_1=2.25,\quad x_2=2.5,\quad x_3=2.75,\quad x_4=3,\quad x_5=3.25,\quad x_6=3.5,\quad x_7=3.75 \] - **Evaluate \(f(x)\) at each:** 1. \(f(2)\): \[ -4 + 12 - 7 = 1 \] 2. \(f(2.25)\): \[ - (2.25)^2 + 6 \times 2.25 - 7 = -5.0625 + 13.5 - 7 = 1.4375 \] 3. \(f(2.5)\): \[ -6.25 + 15 - 7 = 1.75 \] 4. \(f(2.75)\): \[ -7.5625 + 16.5 - 7 = 1.9375 \] 5. \(f(3)\): \[ -9 + 18 - 7 = 2 \] 6. \(f(3.25)\): \[ -10.5625 + 19.5 - 7 = 1.9375 \] 7. \(f(3.5)\): \[ -12.25 + 21 - 7 = 1.75 \] 8. \(f(3.75)\): \[ -14.0625 + 22.5 - 7 = 1.4375 \] - **Sum of function values:** \[ 1 + 1.4375 + 1.75 + 1.9375 + 2 + 1.9375 + 1.75 + 1.4375 = 13.25 \] - **Approximate area:** \[ \boxed{ 0.25 \times 13.25 = 3.3125 } \] --- ### **Step 5: Actual area using the Fundamental Theorem of Calculus** - **Find an antiderivative \(F(x)\) of \(f(x)\):** \[ f(x) = -x^2 + 6x - 7 \] Integral: \[ F(x) = \int (-x^2 + 6x - 7) dx = - \frac{x^3}{3} + 3x^2 - 7x + C \] - **Evaluate \(F(4)\):** \[ F(4) = - \frac{(4)^3}{3} + 3 \times (4)^2 - 7 \times 4 = - \frac{64}{3} + 3 \times 16 - 28 = - \frac{64}{3} + 48 - 28 = - \frac{64}{3} + 20 \] Expressed with common denominator: \[ F(4) = - \frac{64}{3} + \frac{60}{3} = - \frac{4}{3} \] - **Evaluate \(F(2)\):** \[ F(2) = - \frac{8}{3} + 3 \times 4 - 14 = - \frac{8}{3} + 12 - 14 = - \frac{8}{3} - 2 \] Expressed with denominator 3: \[ F(2) = - \frac{8}{3} - \frac{6}{3} = - \frac{14}{3} \] - **Compute the definite integral:** \[ \text{Area} = F(4) - F(2) = \left(- \frac{4}{3}\right) - \left(- \frac{14}{3}\right) = - \frac{4}{3} + \frac{14}{3} = \frac{10}{3} \] - **Final exact area:** \[ \boxed{ \frac{10}{3} \approx 3.3333 } \] --- ### **### **Summary of Results:** | Step | Description | Result | |---------|------------------------------|------------------| | **Exact integral** | \(\int_2^4 (-x^2 + 6x -7) dx\) | \(\boxed{\frac{10}{3} \approx 3.3333}\) | | **n=4 approximation** | Left endpoints sum | **3.25** | | **n=8 approximation** | Left endpoints sum | **3.3125** | --- **Note:** The approximations get closer to the exact value as \(n\) increases. Here, the \(n=8\) estimate (3.3125) is very close to the actual area (about 3.3333). --- If you'd like further steps (right endpoints, midpoints, or graphing), just ask!