Question

In Exercises \(37-42,\) sketch the region \(R\) of integration and switch the order of integration. $$ \int_{1}^{10} \int_{0}^{\ln y} f(x, y) d x d y $$

Short answer

The double integral \(\int_{1}^{10} \int_{0}^{\ln y} f(x, y) dx dy\) with the order of integration switched becomes \(\int_{0}^{2.3} \int_{e^x}^{10} f(x, y) dy dx\).

Step-by-step solution

Sketch the Region of Integration

We start by plotting the given range of y, i.e., \(1 ≤y≤ 10\), and x, i.e., \(0≤ x ≤lny\). This will provide the region R of integration over which the function \(f(x,y)\) is integrated.

Identify New Limits for Switched Order of Integral

From the plot, we can obtain the new limits for switched order of integration. These limits come from the intersection points of the plotted region. In this case, for \(x\) we can observe the limits range from \(0\) to \(2.3\) (approx value of \(\ln 10\)). For \(y\), it is evident that the lower limit is \(y=e^x\), while the upper limit is \(10\) (constant).

Rewrite the Integral with Switched Order

Now we can rewrite the given double integral with the newly defined limits. It becomes \(\int_{0}^{2.3} \int_{e^x}^{10} f(x, y) dy dx\).