Question

When converted to an iterated​ integral, the following double integral is easier to evaluatein one order than the other. Find the best order and evaluate the integral.

∫∫Rx(5+xy)2dA​;
R=​{(x,y): 0≤x≤3, 1≤y≤2​}

Answer 1

The best order for evaluating the given double integral of ∬(5+xy)^2 dA over the region 0≤x≤3, 1≤y≤2 is integrating with respect to y first and then x, due to the polynomial nature of the integrand in y, resulting in straightforward integration steps.

Step-by-step explanation:

The question concerns finding the best order for evaluating a double integral and then performing the actual integration. The given integral is ∬(5+xy)^2 dA over the region R defined by 0≤x≤3, 1≤y≤2. To find the most suitable order of integration, we should consider how the integrand behaves with respect to each variable and the limits of integration.

Looking at the integrand (5 + xy)^2, integrating with respect to y first seems to be the best option because the expression is polynomial in y for each fixed value of x, leading to straightforward integration. Once the integral with respect to y is evaluated, the result will then be a function of x, which we integrate over the interval from 0 to 3.

Let's perform the integration step by step:

1. Integrate (5+xy)^2 with respect to y over the bounds from 1 to 2.
2. Substitute the evaluated integral into the remaining integral over x.
3. Integrate the resulting expression with respect to x from 0 to 3.

The actual integration can be simplified after conducting the first step and integrating with respect to y, yielding a simpler integrand in x, which can then be easily integrated within the specified bounds.