Definite Integrals, Area Between Two Curves, Continuous and Non-Negative Functions on an Interval

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Area between Two Curves

Suppose that and are two continuous and non-negative functions on an interval such that for all that is, the curve does not cross under the curve for .

We want to find the area bounded above by , below by, and on the sides by and .

Area between two Curves

Area between Two Curves

Area between two Curves

Let [Area under] [Area under] ……

Now using the definition for the area bounded by the curve, x-axis and the ordinates and , we have

Area under ……

Similarly, Area under ……

Using equations and in , we get

……

What happens when the function has negative values also? This formula can be extended by translating the curves and upwards until both are above the x-axis.

To do this let- be the minimum value of on (see Fig. 31.a).

Since

translating the curves

Translating the Curves

translating the curves

Now, the functions and are non-negative on (see Fig. b).

It is intuitively clear that the area of a region is unchanged by translation, so the area between and is the same as the area between and.

Thus, [area under ] [area under ] ……

Now using the definitions for the area bounded by the curve , x-axis and the ordinates and , we have

Area under ……

And Area under ……

The equations and give

Which is same as Thus,

If and are continuous functions on the interval , and, then the area of the region bounded above by , below by, on the left by and on the right by is

Square units

If the curves intersect then the sides of the region where the upper and lower curves intersect reduces to a point, rather than a vertical line segment.

Example:

Find the area of the region enclosed between the curves and .

Solution:

We know that is the equation of the parabola which is symmetric about the y-axis and vertex is origin and is the equation of the straight line. (See Fig.).

The equation of the parabola

The Equation of the Parabola

The equation of the parabola

A sketch of the region shows that the lower boundary is and the upper boundary is . These two curves intersect at two points, say A and B. Solving these two equations we get

When and when

The required area

square units

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