1. **Problem Statement:** Find the area under the curve of the function $f(x) = 11x^3 - 8x^2 + 5x + 7$ on the interval $\left[-\frac{3}{4}, \frac{9}{4}\right]$ using Left Hand Rule (LHR), Midpoint Rule (MHR), and Right Hand Rule (RHR) Riemann sums with $n$ sub-intervals. 2. **Formula and Explanation:** - The interval length is $b - a = \frac{9}{4} - \left(-\frac{3}{4}\right) = \frac{12}{4} = 3$. - The width of each sub-interval is $\Delta x = \frac{b - a}{n} = \frac{3}{n}$. - The sub-interval points are $x_i = a + i\Delta x = -\frac{3}{4} + i\frac{3}{n}$ for $i=0,1,2,...,n$. 3. **Left Hand Rule (LHR):** - Use the left endpoints $x_0, x_1, ..., x_{n-1}$. - The sum is $L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x$. 4. **Right Hand Rule (RHR):** - Use the right endpoints $x_1, x_2, ..., x_n$. - The sum is $R_n = \sum_{i=1}^n f(x_i) \Delta x$. 5. **Midpoint Rule (MHR):** - Use midpoints $m_i = \frac{x_{i-1} + x_i}{2}$. - The sum is $M_n = \sum_{i=1}^n f(m_i) \Delta x$. 6. **Expressing the sums:** - $L_n = \sum_{i=0}^{n-1} f\left(-\frac{3}{4} + i\frac{3}{n}\right) \frac{3}{n}$ - $R_n = \sum_{i=1}^n f\left(-\frac{3}{4} + i\frac{3}{n}\right) \frac{3}{n}$ - $M_n = \sum_{i=1}^n f\left(-\frac{3}{4} + \left(i - \frac{1}{2}\right)\frac{3}{n}\right) \frac{3}{n}$ 7. **Function substitution:** - For any $x$, $f(x) = 11x^3 - 8x^2 + 5x + 7$. 8. **Interpretation:** - These sums approximate the definite integral $\int_{-\frac{3}{4}}^{\frac{9}{4}} (11x^3 - 8x^2 + 5x + 7) dx$. - As $n \to \infty$, all sums converge to the exact area. 9. **Exact integral calculation:** - Compute the antiderivative: $$F(x) = \frac{11}{4}x^4 - \frac{8}{3}x^3 + \frac{5}{2}x^2 + 7x$$ - Evaluate: $$\text{Area} = F\left(\frac{9}{4}\right) - F\left(-\frac{3}{4}\right)$$ 10. **Calculate $F\left(\frac{9}{4}\right)$:** $$\frac{11}{4} \left(\frac{9}{4}\right)^4 - \frac{8}{3} \left(\frac{9}{4}\right)^3 + \frac{5}{2} \left(\frac{9}{4}\right)^2 + 7 \left(\frac{9}{4}\right)$$ 11. **Calculate $F\left(-\frac{3}{4}\right)$:** $$\frac{11}{4} \left(-\frac{3}{4}\right)^4 - \frac{8}{3} \left(-\frac{3}{4}\right)^3 + \frac{5}{2} \left(-\frac{3}{4}\right)^2 + 7 \left(-\frac{3}{4}\right)$$ 12. **Final area:** - Subtract the two values to get the exact area under the curve. **Summary:** - LHR, MHR, and RHR sums approximate the area using sums of rectangles with heights from left endpoints, midpoints, and right endpoints respectively. - The exact area is found by evaluating the definite integral using the antiderivative. **Slug:** riemann sums **Subject:** calculus