Area between Two Functions
The area between two functions is equal to the area of the function located above minus the area of the function that lies below.
Examples
1.Calculate the area of the space enclosed by the parabola y = x2 + 2 and the straight line that passes through the points A(−1, 0) and B(1, 4).
The equation of the straight line that passing through AB:
2.Find the area of the figure bounded by the function y = x2 and the lines y = x, x = 0 and x = 2
The points of intersection of the parabola and the straight line y = x.
From x = 0 to x = 1, the straight line is above the parabola.
From x = 1 x = 2, the straight line is below the parabola.
3.Find the area of the plane region bounded by y = ln x, y = 2 and the coordinate axes.
Calculate the intersection point of the curve and the line y = 2.
The area is equal to the area of the rectangle ABC0 minus the area under the curve y = ln x.
The rectangular area is the base times height.
The area under the curve y = ln x is:
4.Find the area of the enclosure limited by the parabola y = 4x − x2 and the tangents to the curve at the points of intersection with the x-axis.
Points of intersection:
The equation of the tangent to the parabola at the point (0, 0):
Equation of the tangent to the parabola at the point (4, 0):
5.Calculate the area bounded by the graphs of the functions y2 = 4x and y = x2.
