Double integrals as volume or area

Double integrals as volume

You are probably familiar that in one-variable calculus, the integral $\int_{a}^{b}$f (x)dx for positive f (x) can be interpreted as the area under the curve f (x) over the interval [a, b].

The integral is the area between the curve f (x) and the x-axis.

In the same way, the double integral $\iint_{D}^{}$f (x, y) dA of positive f (x, y) can be interpreted as the volume under the surface z = f (x, y) over the region D. Imagine that the blue object below is the surface z = f (x, y) floating above the xy-plane. The double integral $\iint_{D}^{}$f (x, y) dA can be interpreted as the volume between the surface z = f (x, y) and the xy-plane, i.e, the "cylinder" above the region D.

You can also see this from the Riemann sum approximating the integral

$$\sum_{{i,j}}^{} \Delta \Delta$$

Each term in the Riemann sum is the volume of a thin box with base $\Delta x \times \Delta y$ and height f (x_ij, y_ij).

Hence, the total Riemann sum approximates the volume under the surface by the volume of a bunch of these thin boxes. In the limit as $\Delta$x,$\Delta$y$\to$ 0, we obtain the total volume under the surface over the region D, i.e., $\iint_{D}^{}$f (x, y) dA.

If f (x, y) > g(x, y) can you see how the double integral

$$\iint_{D}^{}$$

is the volume between the surface z = f (x, y) and the surface z = g(x, y)?

Double integrals as area

In one variable calculus, what happens if you integrate the function f (x) = 1 over the interval [a, b]? You can calculate that

$$\int_{a}^{b} \int_{a}^{b} \big\vert _{a}^{b}$$

The integral of the function f (x) = 1 is just the length of the interval [a, b]. It also happens to be the area of the rectangle of height 1 and length (b - a), but we can interpret it as the length of the interval [a, b].

We can do the same trick for double integrals. We can integrate the function f (x, y) = 1 over the region D. We could interpret the integral as the volume under the function f (x, y) = 1 over D. But the integral of f (x, y) = 1 is also the area of the region D. This can be a nifty way of calculating the area of the region D. If we let A be the area of the region D, we can write this as

$$\iint_{D}^{}$$

Click here for an example of using the double integral to compute area.