1. **State the problem:** We want to approximate the area under the curve $y = f(x) = 3 - x^2$ on the interval $[-1.5, 1.5]$ using a midpoint sum with 3 rectangles. 2. **Formula for midpoint sum:** The midpoint sum with $n$ rectangles on $[a,b]$ is given by $$ M_n = \sum_{i=1}^n f\left(\frac{x_{i-1} + x_i}{2}\right) \Delta x $$ where $\Delta x = \frac{b - a}{n}$ and $x_i = a + i\Delta x$. 3. **Calculate $\Delta x$:** $$ \Delta x = \frac{1.5 - (-1.5)}{3} = \frac{3}{3} = 1 $$ 4. **Determine subintervals:** The subintervals are: - $[-1.5, -0.5]$ - $[-0.5, 0.5]$ - $[0.5, 1.5]$ 5. **Find midpoints:** Midpoints $m_i = \frac{x_{i-1} + x_i}{2}$: - $m_1 = \frac{-1.5 + (-0.5)}{2} = -1$ - $m_2 = \frac{-0.5 + 0.5}{2} = 0$ - $m_3 = \frac{0.5 + 1.5}{2} = 1$ 6. **Evaluate $f$ at midpoints:** - $f(-1) = 3 - (-1)^2 = 3 - 1 = 2$ - $f(0) = 3 - 0^2 = 3$ - $f(1) = 3 - 1^2 = 3 - 1 = 2$ 7. **Calculate midpoint sum:** $$ M_3 = \sum_{i=1}^3 f(m_i) \Delta x = (2 + 3 + 2) \times 1 = 7 $$ **Final answer:** The midpoint sum approximation with 3 rectangles is $$ M_3 = 7 $$