Areas & Arc Lengths (College Board AP® Calculus BC): Exam Questions

Let $f$ and $g$ be the functions defined by $f\left(x\right)=5\ln \left(2x+9\right)$ and $g\left(x\right)=x^6+4x^3$. The graphs of $f$ and $g$, shown in the figure above, intersect at $x=A$ and $x=B$ where $A<0$ and $B>0$.

Find the area of the region enclosed by the graphs of $f$ and $g$.

The graph of the family of functions $y=ax\sqrt{9-x^2}$ for $x\ge 0$, where $a$ is a positive constant, has the general shape shown in the figure above.

Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y=ax\sqrt{9-x^2}$ for $a=8$.

The region $R$ is bounded by the curves with equations $x+y^2=1$ and $x+2y^2=2$. Find the area of $R$.

Let $R$ and $S$ be the regions bounded by the graphs of $y=-\sin \left(\pi x\right)$ and $y=x^3-9x$ between $x=-3$ and $x=3$, as shown below.

Find the sum of the areas of the regions $R$ and $S$, writing your answer in an exact form.

Function $f$ is given by $f\left(x\right)=\frac{x}{x^2+4}$. Find the area of the region bounded by $f\left(x\right)$, the $x$-axis, and the vertical lines $x=-2$ and $x=4$. The graph of $f$ and the the two vertical lines are shown below.

Write your answer in an exact form.

Find the length of the curve $y=f\left(x\right)$ on the interval $0\le x\le 5$ where $f\left(x\right)=10\cos \left(\frac{\pi x}{5}\right)$ .

The shaded region $R$ is bounded by the graphs of the functions $f$ and $g$, where $f\left(x\right)=x^2-2x$ and $g\left(x\right)=x+\text{sin}\left(\pi x\right)$, as shown in the figure.

(Note: Your calculator should be in radian mode.)

Find the area of $R$. Show the setup for your calculations.

Let $f$ and $g$ be the functions defined by $f\left(x\right)=\ln (x+3)$ and $g\left(x\right)=x^4+2x^3$. The graphs of $f$ and $g$, shown in the figure above, intersect at $x=-2$ and $x=B$, where $B>0$.

Find the area of the region enclosed by the graphs of $f$ and $g$.

The functions $f$ and $g$ are defined by $f\left(x\right)=x^2+2$ and $g\left(x\right)=x^2-2x$, as shown in the graph.

Let $R$ be the region bounded by the graphs of $f$ and $g$, from $x=0$ to $x=2$, as shown in the graph. Write, but do not evaluate, an integral expression that gives the area of region $R$.

The graphs of the functions $f$ and $g$ are shown in the figure for $0\le x\le 3$. It is known that $g\left(x\right)=\frac{12}{3+x}$ for $x\ge 0$. The twice-differentiable function $f$, which is not explicitly given, satisfies $f\left(3\right)=2$ and ${\int }_0^{3}f\left(x\right) \text{d}x=10$.

Find the area of the shaded region enclosed by the graphs of $f$ and $g$.

The function $f$ is twice differentiable for all $x$ with $f\left(0\right)=0$. Values of $f\text{'}$, the derivative of $f$, are given in the table for selected values of $x$.

For $x\ge 0$, the function $h$ is defined by $h\left(x\right)={\int }_0^x\sqrt{1+(f\text{'}(t){)}^2} \text{d}t$. Find the value of $h\text{'}\left(\pi \right)$. Show the work that leads to your answer.

What information does ${\int }_0^{\pi }\sqrt{1+(f\text{'}(x){)}^2} \text{d}x$ provide about the graph of $f$?

Let $f$ and $g$ be the functions defined by $f\left(x\right)=2+1.2x+e^{x^2-1.8x}$ and $g\left(x\right)=x^4-6x^2+5x+3$. Let $R$ and $S$ be the two regions enclosed by the graphs of $f$ and $g$ shown in the figure above. Points $a,b$ and $c$ are the intersections of the two graphs.

Find the sum of the areas of regions $R$ and $S$.

Let $R$ be the region in the first quadrant bounded by the $x$-axis and the graphs of $y=\ln \left(3x+1\right)$ and $y=4-2x$, as shown in the figure below.

The horizontal line $y=k$ divides $R$ into two regions of equal area. Write an equation involving one or more integrals whose solution allows the value of $k$ to be found. You do not need to solve the equation.

Find the exact area of the region enclosed by the graph of $y=x\sqrt{1+x}$, the $x$-axis, and the line $x=1$.

Write your answer in the form $a\left(b+c^d\right)$ where $b$ and $c$ are positive integers and $a$ and $d$ are positive rational numbers to be found.

The graph below shows the functions $f\left(x\right)=2\cos \left(2x\right)+2$ and $g\left(x\right)=-2\cos \left(2x\right)$.

Find the area of the shaded region.

The graph below shows the functions $f\left(x\right)=\cos x$ and $g\left(x\right)=\sin 2x$.

One of the points of intersection of $f\left(x\right)$ and $g\left(x\right)$ is at $x=\frac{\pi }{6}$.

Find the ratio of the areas $R_1:R_2$ in the form $a:b$ where $a$ and $b$ are integers.

Find, in terms of $k$, the arc length of the curve $y=\frac{e^x+e^{-x}}{2}$ on the interval $\left[-k,k\right]$ where $k>0$.

A company designs spinning toys using the family of functions $y=cx\sqrt{4-x^2}$, where $c$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $x$-axis and the graph of $y=cx\sqrt{4-x^2}$, for some $c$. Each spinning toy is in the shape of the solid generated when such a region is revolved about the $x$-axis. Both $x$ and $y$ are measured in inches.

Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y=cx\sqrt{4-x^2}$ for $c=6$.

Let $R$ be the region enclosed by the graphs of $g\left(x\right)=-2+3\text{cos}\left(\frac{\pi }{2}x\right)$ and $h\left(x\right)=6-2{\left(x-1\right)}^2$, the $y$-axis, and the vertical line $x=2$, as shown in the figure below.

Find the area of $R$.