Calculus

Area Between Curves

Area by Integration - Problem Solving

         

Let SS be the area of the region bounded by the curve y=x3y=x^3 and the tangent line to the curve at (1,1)(1,1). If S=abS = \frac{a}{b}, where aa and bb are coprime positive integers, what is the value of a+ba+b?

Let S1S_1 be the area of the region bounded by y=x2y=x^2 and y=1y=1. Let S2S_2 be the area of the region bounded by y=x2y=x^2 and y=78x2+k,y=\frac{7}{8} x^2+k, where k>0 k > 0 . If S2=S1S_2=S_1, what is the value of 1k\frac{1}{k}?

Let SS be the area of the region bounded by the xx-axis and the graph of y=k19x18+ky=-k^{19} x^ {18} +k, where k>0k>0. What is the value of 152S152S?

Let S1S_1 be the area of the region bounded by the curves y=x2y=x^2 and y=1y=1. Let S2S_2 be the area of the region bounded by the curves y=x2y=x^2 and y=1mx2+mm1,y=\frac{1}{m}x^2+\frac{m}{m-1}, where mm is a positive integer strictly greater than 1.
If aa and bb are coprime positive integers, and S2S1=ab\frac{S_2}{S_1}=\frac{a}{b}and a+b=925a+b=925, what is mm?

The area bounded by the parabola y=x(6x) y=x(6-x) and the xx-axis is divided into two regions by the line y=xy=x. If S1S_1 is the region above y=xy=x and S2S_2 is the region below y=xy=x, then S1:S2=a:bS_1 : S_2 = a : b, where aa and bb are coprime positive integers. What is the value of a+ba+b?

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