1. **State the problem:** Estimate the area under the curve of the function $f(x) = x^2$ from $x=0$ to $x=1$ using the midpoint rule with first 2 rectangles, then 4 rectangles. 2. **Formula and explanation:** The midpoint rule approximates the area under a curve by dividing the interval into $n$ equal subintervals, then using the function value at the midpoint of each subinterval as the height of a rectangle. The area estimate is: $$\text{Area} \approx \sum_{i=1}^n f(m_i) \Delta x$$ where $\Delta x = \frac{b-a}{n}$ and $m_i = a + \left(i - \frac{1}{2}\right) \Delta x$ is the midpoint of the $i$-th subinterval. 3. **Calculate with 2 rectangles:** - Interval length: $1 - 0 = 1$ - Width of each rectangle: $\Delta x = \frac{1}{2} = 0.5$ - Midpoints: - $m_1 = 0 + (1 - 0.5) \times 0.5 = 0.25$ - $m_2 = 0 + (2 - 0.5) \times 0.5 = 0.75$ - Heights: - $f(0.25) = (0.25)^2 = 0.0625$ - $f(0.75) = (0.75)^2 = 0.5625$ - Area estimate: $$0.5 \times (0.0625 + 0.5625) = 0.5 \times 0.625 = 0.3125$$ 4. **Calculate with 4 rectangles:** - Width of each rectangle: $\Delta x = \frac{1}{4} = 0.25$ - Midpoints: - $m_1 = 0 + (1 - 0.5) \times 0.25 = 0.125$ - $m_2 = 0 + (2 - 0.5) \times 0.25 = 0.375$ - $m_3 = 0 + (3 - 0.5) \times 0.25 = 0.625$ - $m_4 = 0 + (4 - 0.5) \times 0.25 = 0.875$ - Heights: - $f(0.125) = (0.125)^2 = 0.015625$ - $f(0.375) = (0.375)^2 = 0.140625$ - $f(0.625) = (0.625)^2 = 0.390625$ - $f(0.875) = (0.875)^2 = 0.765625$ - Area estimate: $$0.25 \times (0.015625 + 0.140625 + 0.390625 + 0.765625) = 0.25 \times 1.3125 = 0.328125$$ 5. **Final answers:** - Using 2 rectangles, the estimated area is $0.3125$. - Using 4 rectangles, the estimated area is $0.328125$. These estimates approach the exact integral value of $\int_0^1 x^2 dx = \frac{1}{3} \approx 0.3333$ as the number of rectangles increases.