Suppose y=f(x) and y=g(x) are two functions whose graphs...

Suppose $y=f(x)$ and $y=g(x)$ are two functions whose graphs intersect at the three points $(0,4),(2,2)$ and $(4,0)$ with $f(x)>g(x)$ for $0<x<2$ and $f(x)<g(x)$ for $2<x<4 .$ If $\int_{0}^{4}[f(x)-g(x)] d x=10$ and $\int_{2}^{4}[g(x)-f(x)] d x=5$, then the area between two curves for $0<x<2$, is (a) 5 (b) 10 (c) 15 (d) 20

Suppose $y=f(x)$ and $y=g(x)$ are two functions whose graphs intersect at the three points $(0,4),(2,2)$ and $(4,0)$ with $f(x)>g(x)$ for $0<x<2$ and $f(x)<g(x)$ for $2<x<4 .$ If $\int_{0}^{4}[f(x)-g(x)] d x=10$
and $\int_{2}^{4}[g(x)-f(x)] d x=5$, then the area between two curves for $0<x<2$, is
(a) 5
(b) 10
(c) 15
(d) 20

Key Concepts

Definite Integral

A definite integral computes the accumulation of a quantity over a specified interval, often representing areas under curves when interpreted geometrically. It takes into account the sign of the function values, thus requiring adjustments (such as absolute values) when computing an area between curves.

Intersection of Graphs

The points where two graphs intersect are key because they indicate where the functions are equal, and thus, where the roles of the upper and lower functions may switch. These intersection points serve as natural boundaries to break up the overall integration into sections where the function order is consistent.

Piecewise Integration

When the relative positions of two functions change over an interval, the integration must be done piecewise—splitting the overall interval into subintervals on which one function remains consistently above or below the other. This ensures the correct computation of area by accounting for changes in the sign of the difference between the functions.

Area Between Curves

The area between curves is determined by integrating the difference between the two functions over the interval of interest. This concept requires knowing which function lies above the other so that the integrand represents the nonnegative distance between the curves, and then computing the definite integral to obtain the total area.

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