Question

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=x, y=2 x, x=2 $$

Short answer

The area of the region bounded by the graphs of the equations y=x, y=2x, and x=2 is 2 square units.

Step-by-step solution

Identify the bounds

The graph in the x-y plane bounded by the equations y=x, y=2x and x=2 is a triangular region in the first quadrant. The limits of the integral in the x-direction are from the y-x line to the line x=2. In y-direction, the limits are from the line y=x to the line y=2x.

Set up the integral

With the limits defined, set up the double integral to calculate the area. The integrand is 1 as we are finding the area. In doubles integration, the integral is taken first with respect to y, then x. The integral then becomes: \[\int\int dxdy = \int_{0}^{2} \int_{x}^{2x} dy dx\]

Evaluate the inner integral

Perform the inner integral first (with respect to y), which results in evaluating (y) from x to 2x. This gives: \[\int_{0}^{2} [y]_{x}^{2x} dx = \int_{0}^{2} (2x - x) dx = \int_{0}^{2} x dx\]

Evaluate the outer integral

Now perform the outer integral (with respect to x) which gives: \[[x^2/2]_{0}^{2} = 2^2/2 - 0 = 2\]