Double Integrals — Question 2

Question 2 – Solution

Strategy. Express every sample point and area in terms of its indices, then separate the finite sums algebraically before taking the two-parameter limit.

Step 1: Grid and samples We have $$\Delta x=\frac 1m,\qquad \Delta y=\frac 2n,\qquad \Delta A=\frac{2}{mn},$$ and the upper-right sample in cell $(i,j)$ is $(i/m,2j/n)$. Thus $$S_{m,n}=\sum_{i=1}^{m}\sum_{j=1}^{n} \left(\frac{i}{m}+\frac{4j}{n}\right)\frac{2}{mn}.$$

Step 2: Exact finite sum Using $\sum_{k=1}^{N}k=N(N+1)/2$, $$\begin{align*} S_{m,n} &=\frac{2}{m^2n}\sum_{i=1}^{m}\sum_{j=1}^{n}i +\frac{8}{mn^2}\sum_{i=1}^{m}\sum_{j=1}^{n}j\\ &=\frac{2}{m^2n}\left(n\frac{m(m+1)}2\right) +\frac{8}{mn^2}\left(m\frac{n(n+1)}2\right)\\ &=\frac{m+1}{m}+4\frac{n+1}{n}. \end{align*}$$

Step 3: Definition limit Both correction terms vanish independently, so $$\boxed{\iint_R(x+2y)\,dA=\lim_{m,n\to\infty}S_{m,n}=1+4=5}.$$ The answer lies between $0\cdot 2$ and $5\cdot 2=10$, the bounds obtained from the extreme values of $f$ on the area-$2$ rectangle.