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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 y-axis

If the blue curve in the above diagram represents the graph of \(x={y}^3-300y\) and the vertical lines are tangent lines to the curve, what is the area of the shaded region?

What is the area of the region bounded by the curve \(x= -{y}^2+100\) and the \(y\)-axis ?

Let \(S\) be the area bounded by \(y^2 \leq x\), \(x \leq 25\) and \(y \geq 0\). If \(S = \frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?

What is the area of the region bounded by the curve \(x= -6y(y-4)\) and the \(y\)-axis ?

If \(g(x)\) is the inverse function of \(f(x)=10\cos x\) \((0 \leq x \leq \frac{\pi}{2}) ,\) what is the area of the region bounded by \(y=g(x) ,\) the \(x\)-axis and \(y\)-axis?

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