Question

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{1}(3 x+4 y) d y d x $$

Short answer

The region of integration is a rectangle in the xy-plane with corners at (0,0), (0,1), (2,0), and (2,1). The value of the double integral over this region is 10.

Step-by-step solution

Sketching the region of integration

The region of integration is determined by the limits of the integral. Here, \(y\) ranges from 0 to 1, and \(x\) ranges from 0 to 2. This region can be represented graphically as a rectangle in the xy-plane, with corners at the points (0,0), (0,1), (2,0), and (2,1).

Calculating the inner integral

The integrand is \(3x + 4y\), which will first be integrated with respect to \(y\). This gives \( \int_{0}^{1} (3x + 4y) dy = [3xy + 2y^2]_{0}^{1} = 3x + 2 - 0 = 3x + 2 \).

Calculating the outer integral

The result from the previous step, \(3x + 2\), will then be integrated with respect to \(x\). This gives \( \int_{0}^{2} (3x + 2) dx = [\frac{3}{2}x^2 + 2x]_{0}^{2} = \frac{3}{2}*4 + 2*2 - 0 = 6 + 4 = 10 \).

Key concepts

Region of Integration

The region of integration is a fundamental aspect when evaluating multiple integrals. It represents the part of the plane or space over which the integration is performed. Visualizing the region can greatly aid in setting up the correct limits for the integrals. In our exercise, the region of integration is a rectangle in the xy-plane, delineated by the limits of integration for each variable: for y from 0 to 1, and for x from 0 to 2.

Imagine a rectangular plot of land with its corners at coordinates (0,0), (1,0), (0,2), and (1,2); this is the actual area where the function will 'come to life' and where every point within contributes to the total integral. Sketching this region ensures we understand the boundaries and sets us up for accurate integration.

Inner and Outer Integral

When working with double integrals, the concepts of the inner and outer integral are essential. The inner integral refers to the integration carried out with respect to one variable, holding the other variable constant; whereas the outer integral is the second integration, acting over the result of the inner integral, now integrating with respect to the other variable.

In our example, the inner integral was calculated first with respect to y, treating x as a constant. The outer integral was then taken with respect to x, incorporating the result of the inner integral. Chaining these integrals together is how we mesh the two-dimensional information into a single numerical value. The order of these operations is crucial, and flipping the order would mean reassessing our region of integration.

Integration Limits

Integration limits define the boundaries over which the integral is computed for each variable. Accurately setting these limits is critical for getting the correct result. In the context of double integrals, we also refer to these as the limits of integration for the inner and outer integrals.

For our exercise, the inner integral has limits from 0 to 1 for y, and the outer integral has limits from 0 to 2 for x. These limits are based on the region of integration we previously sketched. Ensuring that these limits are set correctly according to the region's boundaries is vital because any mistake here will lead to incorrect results for the integrals.

Mathematical Integration

Mathematical integration is a core concept in calculus, capturing the idea of accumulation or 'adding up' infinitesimal contributions to compute areas, volumes, and other quantities. Double integrals extend this concept to two dimensions, allowing us to add up function values over a two-dimensional region.

Through integration, we essentially break the region down into infinitesimally small rectangles (in the x-y plane), compute the product of the function value at each rectangle's location and the area of these rectangles, and then sum them all up. In our exercise, we integrate the function 3x + 4y over a rectangular region, accumulating the weighted contributions across the plane to reach the final value of 10.