Final answer:
To find the area of the region enclosed by the curves y = x², y = 4x, and x = 0 in the first quadrant, one must sketch the curves, determine the points of intersection, and integrate the difference of the functions over the range of intersection.
Step-by-step explanation:
Calculating the Area of a Region Enclosed by Curves
The question requires sketching the region in the first quadrant enclosed by the equations y = x², y = 4x, and x = 0 and then finding the area of the region. To determine the area, we need to integrate between the points where the curves intersect, which signifies the limits of integration.
Firstly, to sketch the region, plot the given equations in the first quadrant. The equation y = x² is a parabola opening upwards, y = 4x is a straight line with a slope of 4, and x = 0 is the y-axis. The intersection points of y = x² and y = 4x determine the bounds of the region along the x-axis. Set the equations equal to each other and solve for x to find these points.
After sketching the curves, select whether to integrate with respect to x or y. Since the lines and parabola are functions of x, it is simpler to integrate with respect to x. The points of intersection can be found by setting x² = 4x, which yields x = 0 and x = 4. These values become the limits of integration for the integral that will find the area between these curves from x = 0 to x = 4.
To find the region's area, calculate the integral of the upper function minus the lower function within the intersection points. The integral is ∫(4x - x²) dx from x = 0 to x = 4. This integral represents the total area of the region enclosed by the curves and the x-axis between x = 0 and x = 4.