The area of the region bounded by the curve y=f(x), the X...

The area of the region bounded by the curve $y=f(x)$, the $X$ -axis and the lines $x=a$ and $x=b$, where $-\infty<a<b<-2$ is (a) $\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x+b f(b)-a f(a)$ (b) $-\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x+b f(b)-a f(a)$ (c) $\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x-b f(b)+a f(a)$ (d) $-\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x-b f(b)+a f(a)$

The area of the region bounded by the curve $y=f(x)$, the $X$ -axis and the lines $x=a$ and $x=b$, where $-\infty<a<b<-2$ is
(a) $\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x+b f(b)-a f(a)$
(b) $-\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x+b f(b)-a f(a)$
(c) $\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x-b f(b)+a f(a)$
(d) $-\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x-b f(b)+a f(a)$

Key Concepts

Area Under a Curve

The area under a curve is determined by the definite integral of a function over a given interval. However, when the curve lies below the x-axis, the computed integral represents a negative value, necessitating adjustments in the formulation to obtain a positive area, which is the true measure of the region.

Sign Correction in Area Calculation

When a function is negative over an interval, the definite integral gives a signed area that can be negative even though area itself must be nonnegative. Recognizing this requires applying a sign correction, often by taking the absolute value or introducing a negative sign in the integration formula, to ensure that the resulting area calculation reflects the true magnitude of the region.

Definite Integration

Definite integration involves calculating the accumulation of quantities—such as areas under curves—over a closed interval. It provides a means of summing an infinite number of infinitesimally small contributions between two bounds, resulting in a net total that may represent physical quantities like area.

Integration by Parts

Integration by parts is a technique for integrating products of functions by redistributing the differentiation from one function to another based on the product rule of differentiation. This method often produces terms evaluated at the boundaries of the interval, which are crucial in deriving formulas with additional term corrections.

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