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Area Between Curves

In the language of Calculus, the area between two curves is essentially a difference of integrals. The applications of this calculation range from macroeconomic tax models to signal analysis.

Curve and x-axis

Find the area of the region bounded by \(y \geq x^2 - 25 \) and \(y \leq 0\).

What is the area of the region bounded by the curve \(y= 9{x}^2-54x\) and the \(x\)-axis ?

What is area of the region bounded by the curve \(y = \sqrt{x} \) and the \(x\)-axis in the interval \(0 \leq x \leq 4?\)

Parallel<em>AngleBisector

ParallelAngleBisector

Consider the region bounded by the curve \(f(x)= x(x-b)(x-a)\) and the \(x\)-axis, where \(b=5\) and \(a > 5\). If the bounded region above the \(x\)-axis and the bounded region below the \(x\)-axis have the same area, what is the value of \(a ?\)

Let \(C\) be the curve obtained by a parallel translation of the curve \(y=a{x}^2 \) \((a>0)\) by \(4\) units in the positive direction of the \(x\)-axis. If the area of the region bounded by the curve \(y=a{x}^2 ,\) the curve \(C,\) and the \(x\)-axis is \(16,\) what is \(a?\)

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