If the straight line x=b divide the area enclosed by...

If the straight line $x=b$ divide the area enclosed by $y=(1-x)^{2}, y=0$ and $x=0$ into two parts $R_{1}(0 \leq x \leq b)$ and $R_{2}(b \leq x \leq 1)$ such that $R_{1}-R_{2}=\frac{1}{4}$, Then, $b$ equals to $\quad$ [Single Correct Optlon 2011] (a) $\frac{3}{4}$ (b) $\frac{1}{2}$ (c) $\frac{1}{3}$ (d) $\frac{1}{4}$

If the straight line $x=b$ divide the area enclosed by $y=(1-x)^{2}, y=0$ and $x=0$ into two parts $R_{1}(0 \leq x \leq b)$ and $R_{2}(b \leq x \leq 1)$ such that $R_{1}-R_{2}=\frac{1}{4}$, Then, $b$ equals to $\quad$ [Single Correct Optlon 2011]
(a) $\frac{3}{4}$
(b) $\frac{1}{2}$
(c) $\frac{1}{3}$
(d) $\frac{1}{4}$

Key Concepts

Solving Equations Involving Integrals

Once areas have been defined as definite integrals, the next step involves setting up equations based on given relationships between these areas. Solving such equations often combines the evaluation of definite integrals with algebraic manipulation to isolate and determine unknown parameters. This method is frequently necessary in optimization and partitioning problems in calculus.

Partitioning Regions

Partitioning a region refers to dividing a larger area into smaller subregions by introducing additional boundaries, such as vertical or horizontal lines. In problems involving integration, shifting the limits of integration appropriately allows one to separate the overall area into distinct parts, which can then be compared or further analyzed according to specified conditions.

Area Calculation using Integration

The determination of the area enclosed by curves involves setting up an integral where the function, usually representing a curve, is integrated with respect to x (or another variable) over a specified interval. This method effectively sums the infinitesimally small areas (slices) to give the total area, and is especially useful when dealing with functions representing smooth curves.

Definite Integration

Definite integration is the process of calculating the accumulated sum of a function over a specific interval. It allows us to determine areas under curves and is based on the Fundamental Theorem of Calculus. This concept is critical when quantifying regions bounded by a function and the x-axis or between multiple curves over given limits.

Please give Ace some feedback