The area is the definite integral of y = 2x from 0 to 3: Area = the integral from 0 to 3 of 2x dx = [x^2] from 0 to 3 = (3^2) - (0^2) = 9. So the required area is 9 square units. (This agrees with the area of the triangle of base 3 and height 6, which is (1/2)(3)(6) = 9.)

Area = integral(a to b) y dx

The equation x^2 + y^2 = 16 represents a circle centred at the origin with radius r = 4 (since r^2 = 16). The area enclosed by a circle of radius r is pi r^2. So the area = pi (4)^2 = 16 pi square units. (By integration, the area = 4 times the integral from 0 to 4 of sqrt(16 - x^2) dx = 16 pi.)

Area of circle = pi r^2

If a region is symmetric about the x-axis, the y-axis or the origin, we can compute the area of one symmetric part and then multiply by the appropriate factor to get the total area. For example, a circle x^2 + y^2 = a^2 is symmetric about both axes, so its total area equals 4 times the area of the part lying in the first quadrant. Using symmetry reduces the amount of integration needed and simplifies the calculation.