The Method of Exhaustion

Welcome to Calculus IA. Calculus is an important branch of mathematics that studies continuous change and accumulated quantities, and it plays a major role in modern society. Throughout this course we follow the from-integration-to-differentiation design (if you prefer the traditional from-differentiation-to-integration logic, please study the other course, Calculus IB); I hope this course serves you well.

Consider a problem you often meet in daily life: walking across campus or visiting a scenic area, you pass a large lake. Have you ever wondered how large that lake's area actually is? Given adequate tools, how would you measure the area of a completely irregular lake?

One intuitive idea: the areas of regular shapes (rectangles and squares above all) are extremely easy to compute. So we can first throw a huge rectangle around the entire lake. Clearly, the lake's area must be less than or equal to the area of that big rectangle.

Going further, we can draw an evenly spaced grid inside the big rectangle, cutting it into many small squares of identical size (say, with sides of 100 meters). By counting these squares, we can produce a more precise estimate of the irregular lake's area.

Estimate the lake's area by counting grid cells: every small square is identical, with sides of 100 meters and an area of 10,000 square meters. First count only the "fully inside" cells (backslash shading) — they exactly tile the lake's interior, and the area they assemble is the smallest possible value of the lake's area (the lower bound). What is this lower bound? (Enter in units of 10,000 square meters)

Now estimate the largest possible value (the upper bound). This time also count the "boundary-touching" slash-shaded cells — any cell that touches the lake at all (even along a single edge or at a corner) counts. The area assembled from all such cells is an upper bound for the lake's area. What is this upper bound? (Enter in units of 10,000 square meters)

Excellent — you have grasped the core idea of the method of exhaustion. Look for the fully-inside cells (fixing the lower bound) and the boundary-touching cells (fixing the upper bound); through this square-tiling puzzle we have locked an unknown true area inside the interval formed by these two numbers.

But this is still not perfect: the gap that represents the error remains large. A highly insightful idea now appears: if we cut the grid finer (say, sides of 10 meters, or even 1 meter), the range of the estimate grows narrower and narrower.

Moreover, to make the measurement systematic and expressible by a mathematical formula, we need not piece cells together haphazardly. If we fix a uniform scanning direction — left to right, one column at a time — you will find that the small squares stacked vertically merge perfectly into rows of "vertical strip rectangles" of varying heights.

Once the two-dimensional grid evolves into rows of adjacent vertical rectangles, we are one step away from the truth of calculus. How can these ever-narrowing vertical rectangles yield an absolutely exact area? That is the ultimate question the next lesson explores.