적분법 전체 문제

The data in the following table are used to estimate the average power output produced by Peter Sagan for each of the last 18 sec of Stage 1 of the 2012 Tour de France.

Average Power Output

Source: sportsexercisengineering.com

Estimate the net energy used in kilojoules (kJ), noting that 1W = 1 J/s, and the average power output by Sagan during this time interval.

Let ${d}_{j}$ denote the hours of daylight and ${\delta }_{j}$ denote the increase in the hours of daylight from day $j-1$ to day j in Fargo, North Dakota, on the jth day of the year. Interpret ${d}_{1}+\sum_{j=2}^{365}{\delta }_{j}.$

[T] $y={x}^{4}-5{x}^{2}+4$ on the interval $[-2,2],$ which has an exact area of $\frac{32}{15}$

[T] $y=\ln x$ on the interval $[1,2],$ which has an exact area of $2\ln (2)-1$

The area of a semicircle of radius 1 can be expressed as ${\int }_{-1}^{1}\sqrt{1-{x}^{2}}dx.$ Use the substitution $x=\cos t$ to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral.

The area of the top half of an ellipse with a major axis that is the x-axis from $x=-a$ to $x=a$ and with a minor axis that is the y-axis from $y=-b$ to $y=b$ can be written as ${\int }_{-a}^{a}b\sqrt{1-\frac{{x}^{2}}{{a}^{2}}}dx.$ Use the substitution $x=a\cos t$ to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.

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] $x={e}^{y}\,\text{and}\,y=x-2$

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

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] $y=3{x}^{2}+8x+9\,\text{and}\,3y=x+24$

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] ${x}^{2}={y}^{3}\,\text{and}\,x=3y$

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

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.

Derive the formula for the volume of a sphere using the slicing method.

Use the slicing method to derive the formula for the volume of a cone.