넓이(여러 가지 함수)

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y={\sin }^{3}x+2,y=\tan \,x,x=-1.5,\,\text{and}\,x=1.5$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y=\sqrt{1-{x}^{2}}\,\text{and}\,{y}^{2}={x}^{2}$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y=\sqrt{1-{x}^{2}}\,\text{and}\,y={x}^{2}+2x+1$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $x=4-{y}^{2}\,\text{and}\,x=1+3y+{y}^{2}$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y=\cos \,x,y={e}^{x},x=-\pi ,\,\text{and}\,x=0$

The largest triangle with a base on the $x\text{-axis}$ that fits inside the upper half of the unit circle ${y}^{2}+{x}^{2}=1$ is given by $y=1+x$ and $y=1-x.$ See the following figure. What is the area inside the semicircle but outside the triangle?

An amusement park has a marginal cost function $C(x)=1000{e}^{-x}+5,$ where $x$ represents the number of tickets sold, and a marginal revenue function given by $R(x)=60-0.1x.$ Find the total profit generated when selling $550$ tickets. Use a calculator to determine intersection points, if necessary, to two decimal places.

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function $H(t)=1-\cos ((\pi t)/2)$ whereas the speed of the tortoise is $T(t) = 0.1032t$ where $t$ is time measured in hours and the speed is measured in miles per hour. Find the area between the curves from time $t=0$ to the first time after one hour when the tortoise and hare are traveling at the same speed. What does it represent? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function $H(t)=(1/2)-(1/2)\cos (2\pi t)$ whereas the speed of the tortoise is $T(t)=\sqrt{t},$ where $t$ is time measured in hours and speed is measured in kilometers per hour. If the race is over in $1$ hour, who won the race and by how much? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

For the following exercises, find the area between the curves by integrating with respect to $x$ and then with respect to $y.$ Is one method easier than the other? Do you obtain the same answer?

$y={x}^{2}+2x+1\,\text{and}\,y=-{x}^{2}-3x+4$

For the following exercises, find the area between the curves by integrating with respect to $x$ and then with respect to $y.$ Is one method easier than the other? Do you obtain the same answer?

$y={x}^{4}\text{and}\,x={y}^{5}$

Determine the equations for the sides of the square that touches the unit circle on all four sides, as seen in the following figure. Find the area between the perimeter of this square and the unit circle. Is there another way to solve this without using calculus?

The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.

If the force is constant, the amount of work to move an object from $x=a$ to $x=b$ is $F(b-a).$

The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

If the half-life of $\text{seaborgium-}266$ is $360$ ms, then $k=(\ln (2))/360.$

The volume that has a base of the ellipse ${x}^{2}/4+{y}^{2}/9=1$ and cross-sections of an equilateral triangle perpendicular to the $y\text{-axis}.$ Use the method of slicing.

The region bounded by the curve $y = {x}^{2} - x$ and the x-axis from $x=1\,\text{to}\,x=4,$ rotated around the y-axis using the washer method

$x={y}^{2}$ and $x=3y$ rotated around the y-axis using the washer method

For the following exercises, use the requested method to determine the volume of the solid.

$x=2{y}^{2}-{y}^{3},x=0,\,\text{and}\,y=0$ rotated around the x-axis using cylindrical shells