Understanding Double Integrals

When finding the volume under a surface over a rectangular region, we divide the region into smaller rectangles. For a rectangle bounded by x-values from a to b and y-values from c to d, we cut the x-axis into m parts and y-axis into n parts.

Each small rectangle has area Δx·Δy, and we pick a point $x₁*, y₁*$ in each rectangle. The height of the surface at this point is f$x₁*, y₁*$, giving us a rectangular solid with volume f$x₁*, y₁*$·Δx·Δy. The total volume approximation becomes the sum of all these small volumes.

To find the exact volume, we take the limit as m and n approach infinity, turning our approximation into a double integral: $$V = \int_{c}^{d} \int_{a}^{b} f(x,y) dx dy = \int_{a}^{b} \int_{c}^{d} f(x,y) dy dx$$

Remember this! For functions that can be separated like f(x,y) = g(x)·h(y), a double integral equals the product of two single integrals: $\int_{a}^{b} \int_{c}^{d} f(x,y) dx dy = [\int_{a}^{b} g(x) dx][\int_{c}^{d} h(y) dy]$

Evaluating Double Integrals

When evaluating double integrals, we work from the inside out. First solve the inner integral while treating the other variable as a constant, then solve the resulting outer integral.

For example, with $\int_0^2 \int_0^{\sqrt{x}} \frac{1}{\sqrt{x}} dy dx$, we first integrate with respect to y while treating x as constant. After evaluating the inner integral, we're left with $\int_0^2 \sqrt{x} dx$, which equals $\frac{2}{3}x^{3/2}$ evaluated from 0 to 2.

Sometimes integration techniques like integration by parts are needed. When evaluating $\int_0^1 \int_0^x xe^y dy dx$, after handling the inner integral, we use u-substitution or integration by parts for the resulting $\int_0^1 xe^x dx$.

Pro tip: When you see a double integral with variables in the limits, pay careful attention to which variable you're integrating first. The inner integral's limits may contain the outer variable, but never the other way around!

Double Integrals Over Non-Rectangular Regions

For non-rectangular regions, we need to determine how variables change with respect to each other. There are two approaches to setting up these integrals:

1. y-oriented setup: $\int_c^d \int_{g_1(y)}^{g_2(y)} f(x,y) dx dy$ - Here we fix y values and determine how x varies for each y.

2. x-oriented setup: $\int_a^b \int_{h_1(x)}^{h_2(x)} f(x,y) dy dx$ - Here we fix x values and determine how y varies for each x.

For a region bounded by curves like y = 2x and y = x², we identify intersection points (0,0) and (2,4), then decide which orientation works better. Both setups produce the same result, as shown by the calculation $\int_{y=0}^{y=4} \int_{x=\frac{y}{2}}^{x=\sqrt{y}} x^2y dx dy = \int_{x=0}^{x=2} \int_{y=x^2}^{y=2x} x^2y dy dx = \frac{128}{35}$.

Visualization helps! Sketch the region and boundary curves before setting up your integral. This makes it much easier to determine the correct limits of integration and avoid mistakes.