Suppose that a nonnegative function y=f(x) has a continuous...

Suppose that a nonnegative function $y=f(x)$ has a continuous first derivative on $[a, b]$. Let $C$ be the boundary of the region in the $x y$-plane that is bounded below by the $x$-axis, above by the graph of $f$, and on the sides by the lines $x=a$ and $x=b$. Show that $$ \int_a^b f(x) d x=-\oint_C y d x $$

Key Concepts

Parametrization of Curves

Parametrization involves expressing the coordinates of points on a curve as functions of a single parameter. This method facilitates the computation of line integrals by transforming the integral over a curve into a standard integral over an interval of the parameter. In the context of verifying integral identities, parametrization helps in correctly accounting for the differential elements, such as dx, along different segments of the boundary.

Line Integrals

A line integral extends the notion of a standard definite integral to functions defined along a curve. It computes the accumulation of a function over a path by integrating with respect to an infinitesimal element (like dx) along the curve. The concept is central to relating integrals over paths or curves with corresponding functions defined along those curves.

Curve Orientation

The orientation of a curve indicates the direction in which the curve is traversed. For closed curves, this direction is crucial because reversing the direction of traversal reverses the sign of the line integral. Correct orientation ensures that integrals over multiple segments add appropriately, especially when relating integrals over the boundary of a region to integrals over its interior.

Decomposition of a Boundary Curve

A complex closed boundary can be split into simpler segments that are easier to analyze and integrate over individually. By breaking up the boundary into pieces—such as the top and bottom portions—the integration can be carried out along each segment separately. The combination of these integrals, with attention to their orientation, leads to the desired equality or relation.

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