Double Integrals — Question 1

Question 1 – Solution

Strategy. Translate the partition into four base areas, evaluate the height at each midpoint, and add the resulting prism volumes.

Step 1: Partition data Here $\Delta x=1$, $\Delta y=1/2$, and $$\Delta A=\Delta x\Delta y=\frac 12.$$ The midpoints are $$\left(\frac 12,\frac 14\right),\ \left(\frac 32,\frac 14\right),\ \left(\frac 12,\frac 34\right),\ \left(\frac 32,\frac 34\right).$$

Step 2: Heights and sum In the same order, the four function values are $$\frac{11}{2},\quad \frac 72,\quad 5,\quad 3.$$ Therefore $$S=\frac 12\left(\frac{11}{2}+\frac 72+5+3\right)=\boxed{\frac{17}{2}}.$$

Step 3: Interpretation and bounds The sum is the total volume of four midpoint prisms. On $R$, $f$ decreases as either $x$ or $y$ increases, so $$1=f(2,1)\le f(x,y)\le f(0,0)=6.$$ Since $\operatorname{area}(R)=2$, every Riemann-sum estimate lies between $2$ and $12$; indeed $17/2$ does. This also verifies that every sampled height was positive and on the correct scale.