Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{1}^{2} \int_{2}^{4} d x d y $$

Step-by-Step Solution

Answer

The area of the region R, determined by the double integral, is 2. Changing the order of integration did not result in a different area, demonstrating the integral equivalence.

Step 1: Sketch the Region

The double integral \(\int_{1}^{2} \int_{2}^{4} d x d y\) covers a rectangular region in the xy-plane. We can visualize this as a rectangle with corners at (2,1), (2,2), (4,1), and (4,2)

Step 2: Compute the Double Integral with Original Order

The current order of integration is dx dy. Sequentially perform the integration, first integrate with respect to x from 2 to 4, then integrate with respect to y from 1 to 2. This yields the area \( \int_{1}^{2} \int_{2}^{4} d x d y = (4-2)*(2-1) = 2 \)

Step 3: Change the Order of Integration

The order dx dy can be switched to dy dx. The new double integral will be \(\int_{2}^{4} \int_{1}^{2} d y d x\)

Step 4: Compute the Double Integral with New Order

Now repeat the computation of the area with the new order. This means we first integrate with respect to y from 1 to 2, then with respect to x from 2 to 4. This gives \(\int_{2}^{4} \int_{1}^{2} d y d x = (2-1)*(4-2) = 2 \)

Step 5: Comparison of the results

You will notice that the area obtained in both cases is 2, which demonstrates that changing the order of integration did not affect the result.