Calculus

# Area by Integration - Problem Solving

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 $a+b$?

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