If the area bounded between X -axis and the graph of y=6 x-3...

If the area bounded between $X$ -axis and the graph of $y=6 x-3 x^{2}$ between the ordinates $x=1$ and $x=a$ is 19 sq units, then ' $a$ ' can take the value (a) 4 or $-2$ (b) two values are in $(2,3)$ and one in $(-1,0)$ (c) two values one in $(3,4)$ and one in $(-2,-1)$ (d) None of the above

If the area bounded between $X$ -axis and the graph of $y=6 x-3 x^{2}$ between the ordinates $x=1$ and $x=a$ is 19 sq units, then ' $a$ ' can take the value
(a) 4 or $-2$
(b) two values are in $(2,3)$ and one in $(-1,0)$
(c) two values one in $(3,4)$ and one in $(-2,-1)$
(d) None of the above

Key Concepts

Definite Integration

Definite integration is a fundamental concept in calculus that computes the net area under a curve between two points. It involves calculating the limit of Riemann sums to find the exact accumulation of quantities, often representing areas, displacement, or total change over an interval.

Area Between a Curve and an Axis

This concept involves calculating the region bounded by a function and an axis, typically using integration. When the function crosses the axis, one may need to take absolute values to account for parts of the area below the axis, since the integral computes signed areas.

Solving Equations from Integral Expressions

In many problems, the result of an integral is set equal to a given quantity, leading to an equation where an unknown parameter must be determined. This requires not only performing the integration but also manipulating and solving the resulting equation, which may involve algebraic techniques such as solving quadratic equations.

Quadratic Functions

Quadratic functions, defined by expressions of the form ax² + bx + c, are common in calculus problems. Their graphs are parabolas, which can intersect the axes and have regions to integrate over. Analyzing the roots and behavior of quadratic functions is crucial in determining limits of integration and understanding the geometry of the problem.

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