Outer integral

Definition

The outer integral refers to the integral that is evaluated first in a double integral setup, especially when integrating over non-rectangular regions. It typically represents the integration with respect to the variable that defines the outermost layer or boundary of the region of integration, allowing for the evaluation of areas or volumes in a stepwise manner.

5 Must Know Facts For Your Next Test

1. In a double integral expressed as $$ ext{∬_D f(x,y) dy dx}$$, the outer integral would be the one associated with the dx term when integrating with respect to x first.
2. The limits of integration for the outer integral depend on the projection of the region onto the axis corresponding to that variable.
3. When dealing with non-rectangular regions, it's crucial to carefully analyze how to set up the outer integral to ensure that all parts of the region are covered.
4. Changing the order of integration may require re-evaluating the limits of both the outer and inner integrals based on how the region is defined.
5. The process of evaluating an outer integral often involves finding antiderivatives and substituting limits, which contributes to understanding how quantities accumulate over regions.

Review Questions

• How do you identify which integral is the outer integral when setting up a double integral?
  • To identify the outer integral in a double integral, look at the order of integration specified in the expression. The outer integral corresponds to the variable that is integrated first and typically represents the broader range over which you are summing. For instance, in $$ ext{∬_D f(x,y) dy dx}$$, 'dx' indicates that x is integrated last, making 'dy' the outer integral. It’s also essential to visualize or sketch the region of integration to determine which variable bounds define it.
• Discuss how changing the order of integration affects the evaluation of an outer integral.
  • Changing the order of integration can significantly alter both the limits and complexity of evaluating an outer integral. When switching from an order like $$dy dx$$ to $$dx dy$$, you need to reassess how the region of integration appears on both axes. This reassessment may lead to new limits for both integrals. By analyzing how boundaries shift with this change, it can simplify calculations or provide insight into symmetry or behavior of functions within the region.
• Evaluate a specific example where you calculate a double integral and highlight your approach to finding the outer integral.
  • Consider calculating the double integral $$ ext{∬_D (x+y) dA}$$ over a triangular region bounded by points (0,0), (1,0), and (0,1). We can set up our integrals as $$ ext{∫_0^1 ∫_0^{1-x} (x+y) dy dx}$$. Here, 'dx' is our outer integral because we integrate with respect to x first. As we evaluate this, we compute the inner integral with respect to y first, treating x as constant. Once we find that result, we then integrate with respect to x for our outer integral, ultimately determining our overall accumulation over that triangular area.

Related terms

double integral:

A double integral is an integral that computes the accumulation of quantities over a two-dimensional area, represented as $$ ext{∬_D f(x,y) dA}$$ where D is the region of integration.

iterated integral: An iterated integral is a way to compute double integrals by evaluating one integral at a time, where the outer integral is evaluated before the inner integral.

region of integration:

The region of integration is the specific area over which a double integral is calculated, which can be rectangular or non-rectangular in shape.