91Ó°ÊÓ

When dealing with double integrals, particularly iterated integrals, the process involves integrating a function over a two-dimensional region. The key is to perform the integration one variable at a time.

• Start with the inner integral, integrating with respect to one variable while treating the other as a constant.
• After integrating the inner variable over its respective bounds, move on to the outer integral, addressing the remaining variable.

In our exercise, the iterated integrals are written as \( \int_{1}^{2} \int_{-u^{2}-1}^{-u}(8 u v) \ dv \ du \). To solve this, the procedure is straightforward:- First, evaluate \( \int_{-u^{2}-1}^{-u}(8 u v) \ dv \), keeping \(u\) constant.- After obtaining a result for the inner integral, move to the outer integral \( \int_{1}^{2} \ ... \ du \) and integrate the expression with respect to \( u \).Understanding each level of integration separately simplifies the complex task, making iterated integrals manageable.

Finding an antiderivative is a fundamental step in calculus, particularly within integral calculations. An antiderivative is a function that, when differentiated, yields the original function you started with.

• For the inner integration step \( \int_{-u^{2}-1}^{-u}(8 u v) \ dv \), we find the antiderivative of \( 8uv \) with respect to \( v \).
• This antiderivative results in \( 4uv^2 \).

Finding the antiderivative isn't the end; one must evaluate it at the given bounds. Once simplified, it allows you to determine the initial expression's accumulated value over that interval. For iterated integrals, this approach is essential twice—once for each variable individually treated in the two-tier process.

Integration limits define the range over which the integrals are evaluated. They are crucial in accurately capturing the area or volume under study within the function.

• The inner integral limits, \(-u^{2}-1\) to \(-u\), govern the variable \(v\). These boundaries may vary with respect to the other variable, \(u\), indicating that the integration range changes as you move along the second variable.
• For the outer integral, the fixed bounds from 1 to 2 describe where \( u \) should be evaluated.

Handling integration limits appropriately ensures that calculations remain within the desired region. For our example, once the antiderivatives are found, these numeric limits are substituted into the expressions, leading to the computed numeric result. They translate an abstract integral expression into a meaningful numerical solution.