Statement I Area bounded by y=x(x-1) and y=x(1-x) is (1)/(3)...

Statement I Area bounded by $y=x(x-1)$ and $y=x(1-x)$ is $\frac{1}{3}$
Statement II Area bounded by $y=f(x)$ and $y=g(x)$ is $\left|\int_{a}^{b}(f(x)-g(x)) d x\right|$ is true when $f(x)$ and $g(x)$ lies above $X$ -axis. (Where $a$ and $b$ are intersection of $y=f(x)$ and $y=g(x))$.

Key Concepts

Absolute Value Application in Integration

When computing areas using definite integrals, the result may be negative if the order of subtraction is reversed. The absolute value operation is utilized to rectify this by ensuring that the computed area is non-negative, which is crucial since area, as a measure, cannot be negative. This is particularly important when both functions lie above the x-axis and their order might vary over the interval of interest.

Area Between Curves

This concept involves calculating the region enclosed between two curves by integrating the difference of the functions over the interval where they intersect. This technique requires identifying which function is on top of the other within the interval of integration so that the difference is non-negative, or alternatively, the absolute value is taken to guarantee a positive area measurement.

Definite Integration

Definite integration is a fundamental tool in calculus used to calculate the accumulation of quantities, such as area under a curve, over a specified interval. When applied to areas between curves, it involves integrating the difference of the function values at corresponding points, which represents the vertical distance between them, over the domain defined by their intersection points.

Intersection Points of Functions

Intersection points are the values at which two functions have the same output, and they serve as the limits of integration when determining the area bounded by the curves. Identifying these points accurately is essential, as they determine the interval over which the integration is performed and ensure that the area is correctly computed.

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