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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.
Let \(S\) be the area of the region bounded by the curve \(y=x^3\) and the tangent line to the curve at \((1,1)\). If \(S = \frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?
Let \(S_1\) be the area of the region bounded by \(y=x^2\) and \(y=1\). Let \(S_2\) be the area of the region bounded by \(y=x^2\) and \(y=\frac{7}{8} x^2+k,\) where \( k > 0 \). If \(S_2=S_1\), what is the value of \(\frac{1}{k}\)?
Let \(S\) be the area of the region bounded by the \(x\)-axis and the graph of \(y=-k^{19} x^ {18} +k\), where \(k>0\). What is the value of \(152S\)?
Let \(S_1\) be the area of the region bounded by the curves \(y=x^2\) and \(y=1\). Let \(S_2\) be the area of the region bounded by the curves \(y=x^2\) and \[y=\frac{1}{m}x^2+\frac{m}{m-1},\] where \(m\) is a positive integer strictly greater than 1.
If \(a\) and \(b\) are coprime positive integers, and \(\frac{S_2}{S_1}=\frac{a}{b}\)and \(a+b=925\), what is \(m\)?
The area bounded by the parabola \( y=x(6-x)\) and the \(x\)-axis is divided into two regions by the line \(y=x\). If \(S_1\) is the region above \(y=x\) and \(S_2\) is the region below \(y=x\), then \(S_1 : S_2 = a : b\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?