?Water flows into and out of a storage tank. A graph of the rate of change r(t) of the

Verify by differentiation that the formula is correct. \(\int \ln x d x=x \ln x-x+C\) Equation Transcription: ? Text Transcription: integral ln x dx=x ln x-x+c

Verify by differentiation that the formula is correct. \(\int \tan ^{2} x d x=\tan x-x+C\) Equation Transcription: ? Text Transcription: integral tan^2 x dx=tan x-x+c

Verify by differentiation that the formula is correct. \(\int \frac{1}{x^{2} \sqrt{1+x^{2}}} d x=-\frac{\sqrt{1+x^{2}}}{x}+C\)

Verify by differentiation that the formula is correct. \(\int x \sqrt{a+b x} d x=\frac{2}{15 b^{2}}(3 b x-2 a)(a+b x)^{3 / 2}+C\)

Find the general indefinite integral. \(int\left(3 x^{2}+4 x+1\right) d x\) ________________ Equation Transcription: Text Transcription: int(3 x^{2}+4 x+1) dx

Find the general indefinite integral. \(\int(5+2 \sqrt{x}) d x\) ________________ Equation Transcription: Text Transcription: int(5+2 \sqrt{x}) dx

Find the general indefinite integral. \(\int(x+\cos x) d x\) Equation Transcription: ? Text Transcription: integral (x+cos x) dx

Find the general indefinite integral. \(\int\left(\sqrt[3]{x}+\frac{1}{\sqrt[3]{x}}\right) d x\) ________________ Equation Transcription: Text Transcription: int(sqrt[3]{x} + frac{1}{sqrt[3]{x}}) dx

Find the general indefinite integral. \(\int\left(x^{1.3}+7 x^{2.5}\right) d x\)

Find the general indefinite integral. \(\int \sqrt[4]{x^{5}} d x\) ________________ Equation Transcription: Text Transcription: int sqrt[4]{x^{5}} dx

Find the general indefinite integral. \(\int\left(5+\frac{2}{3} x^{2}+\frac{3}{4} x^{3}\right) d x\) ________________ Equation Transcription: Text Transcription: int(5 + frac{2}{3} x^{2} + frac{3}{4} x^{3}) dx

Find the general indefinite integral. \(\int\left(u^{6}-2 u^{5}-u^{3}+\frac{2}{7}\right) d u\) ________________ Equation Transcription: Text Transcription: int(u^{6} - 2 u^{5}-u^{3} + frac{2}{7}) du

Find the general indefinite integral. \(\int(u+4)(2 u+1) d u\) ________________ Equation Transcription: Text Transcription: int(u+4)(2 u+1) du

Find the general indefinite integral. \(\int \sqrt{t}\left(t^{2}+3 t+2\right) d t\) ________________ Equation Transcription: Text Transcription: int sqrt{t}(t^{2}+3 t+2) dt

Find the general indefinite integral. \(\int \frac{1+\sqrt{x}+x}{x} d x\) ________________ Equation Transcription: Text Transcription: int frac{1 + sqrt{x}+x}{x} dx

Find the general indefinite integral. \(\int\left(x^{2}+1+\frac{1}{x^{2}+1}\right) d x\) Equation Transcription: Text Transcription: int(x^{2} + 1 + frac{1}{x^{2} + 1}) dx

Find the general indefinite integral. \(\int\left(e^{x}+\frac{1}{x}\right) d x\)

Find the general indefinite integral. \(\int\left(2+3^{x}\right) d x\)

Find the general indefinite integral.

Find the general indefinite integral.

Find the general indefinite integral. \(\int\left(2+\tan ^{2} \theta\right) d \theta\)

Find the general indefinite integral.

Find the general indefinite integral.

Find the general indefinite integral.

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen.

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. \(\int\left(e^{x}-2 x^{2}\right) d x\)

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral. \(\int_{0}^{\pi / 2}(\sqrt{t}-3 \cos t) d t\) ________________ Equation Transcription: Text Transcription: int_{0}^{pi/2}(sqrt{t} - 3 cos t) dt

Evaluate the definite integral. \(\int_{0}^{1} x(\sqrt[3]{x}+\sqrt[4]{x}) d x\) ________________ Equation Transcription: Text Transcription: int_{0}^{1} x(sqrt[3]{x}+\sqrt[4]{x}) dx

Evaluate the definite integral. \(\int_{1}^{4} \frac{\sqrt{y}-y}{y^{2}} d y\) ________________ Equation Transcription: Text Transcription: int_{1}^{4} frac{\sqrt{y} - y}{y^{2}} dy

Evaluate the definite integral. \(\int_{1}^{2}\left(\frac{x}{2}-\frac{2}{x}\right) d x\) ________________ Equation Transcription: Text Transcription: int_{1}^{2}(frac{x}{2} - frac{2}{x}) dx

Evaluate the definite integral. \(\int_{0}^{1}\left(5 x-5^{x}\right) d x\) ________________ Equation Transcription: Text Transcription: int_{0}^{1}(5 x-5^{x}\right) dx

Evaluate the definite integral. \(\int_{-2}^{2}(\sinh x+\cosh x) d x\)

Evaluate the definite integral. \(\int_{0}^{\pi / 4}\left(3 e^{x}-4 \sec x \tan x\right) d x\) ________________ Equation Transcription: Text Transcription: int_{0}^{pi/4} (3e^{x}-4 sec x tan x) dx

Evaluate the definite integral. \(\int_{0}^{\pi / 4} \frac{1+\cos ^{2} \theta}{\cos ^{2} \theta} d \theta\) ________________ Equation Transcription: Text Transcription: int_{0}^{pi/4} frac{1 + cos ^{2} theta}{cos ^{2} theta} d theta

Evaluate the definite integral. \(\int_{0}^{\pi / 3} \frac{\sin \theta+\sin \theta \tan ^{2} \theta}{\sec ^{2} \theta} d \theta\) ________________ Equation Transcription: Text Transcription: int_{0}^{pi/3} frac{sin theta + sin theta tan ^{2} theta}{sec ^{2} theta} d theta

Evaluate the definite integral. \(\int_{3}^{4} \sqrt{\frac{3}{x}} d x\) ________________ Equation Transcription: Text Transcription: integral_3^4 square root 3/x dx

Evaluate the definite integral. \(\int_{-10}^{10} \frac{2 e^{x}}{\sinh x+\cosh x} d x\) ________________ Equation Transcription: Text Transcription: integral_-10^10 2e^x/sinh x+cosh x dx

Evaluate the definite integral. \(\int_{0}^{\sqrt{3} / 2} \frac{d r}{\sqrt{1-r^{2}}}\) Equation Transcription: Text Transcription: integral_0^square root 3/2 dr/square root1-r^2

Evaluate the definite integral. \(\int_{\pi / 6}^{\pi / 2} \csc t\ \cot t\ d t\) Equation Transcription: Text Transcription: integral_pi/6^pi/2 csc t cot t dt

Evaluate the definite integral. \(\int_{0}^{1 / \sqrt{3}} \frac{t^{2}-1}{t^{4}-1} d t\)

Evaluate the definite integral. \(\int_{0}^{2}|2 x-1| d x\) Equation Transcription: Text Transcription: integral_0^2 |2x-1|dx

Evaluate the definite integral. \(\int_{-1}^{2}(x-2|x|) d x\) ________________ Equation Transcription: Text Transcription: integral_-1^2 (x-2|x|)dx

Evaluate the definite integral. \(\int_{0}^{3 \pi / 2}|\sin x| d x\) Equation Transcription: Text Transcription: integral_0^3 pi/2 |sin x|dx

Use a graph to estimate the x-intercepts of the curve \(y=1-2 x-5 x^{4}\). Then use this information to estimate the area of the region that lies under the curve and above the x-axis.

Repeat Exercise 55 for the curve \(y=\left(x^{2}+1\right)^{-1}-x^{4}\). Equation Transcription: Text Transcription: y = (x+1d)^-1 -x^4

The area of the region that lies to the right of the \(y\)-axis and to the left of the parabola \(x=2 y-y^{2}\) (the shaded region in the figure) is given by the integral \(\int_{0}^{2}\left(2 y-y^{2}\right) d y\). (Turn your head clockwise and think of the region as lying below the curve \(x=2 y-y^{2}\) from \(y=0\) to \(y=2\).) Find the area of the region. Equation Transcription: Text Transcription: y x = 2y-y^2 integral_0^2 (2y-y^2)dy y=0 y=2

The boundaries of the shaded region are the \(y\)-axis, the line \(y=1\), and the curve \(y=\sqrt[4]{x}\). Find the area of this region by writing \(x\) as a function of \(y\) and integrating with respect to \(y\) (as in Exercise 57). Equation Transcription: Text Transcription: y = 1 y x y=4th root x

If \(w^{\prime}(t)\) is the rate of growth of a child in pounds per year, what does \(\int_5^{10} w^{\prime}(t) d t\) represent?

The current in a wire is defined as the derivative of the charge: \(I(t)=Q^{\prime}(t)\). (See Example 3.7.3.) What does \(\int_{a}^{b} I(t) d t\) represent? Equation Transcription: Text Transcription: I(????)=????'(????) integral_a^b I(t)

If oil leaks from a tank at a rate of \(r(t)\) gallons per minute at time \(t\), what does \(\int_{0}^{120} r(t) d t\) represent? Equation Transcription: Text Transcription: r(t) integral_0^120 r(t)dt

A honeybee population starts with 100 bees and increases at a rate of \(n^{\prime}(t)\) bees per week. What does \(100+\int_{0}^{15} n^{\prime}(t) d t\) represent?

In Section 4.7 we defined the marginal revenue function \(R^{\prime}(x)\) as the derivative of the revenue function \(R(x)\), where \(x\) is the number of units sold. What does \(\int_{1000}^{5000} R^{\prime}(x) d x\) represent? Equation Transcription: Text Transcription: R'(x) integral_1000^5000 R'(x)dx

If \(f(x)\) is the slope of a trail at a distance of \(x\) miles from the start of the trail, what does \(\int_{3}^{5} f(x) d x\) represent? Equation Transcription: Text Transcription: f(x) integral_3^5 f(x) dx

If \(h(t)\) is a person's heart rate in beats per minute \(t\) minutes into an exercise session, what does \(\int_{0}^{30} h(t) d t\) represent? ________________ Equation Transcription: Text Transcription: h(t) integral_0^30 h(t)dt

If the units for x are feet and the units for a(x) are pounds per foot, what are the units for da/dx ? What units does \(\int_{2}^{8} a(x) d x\) have?

If x is measured in meters and f(x) is measured in newtons, what are the units for \(\int_{0}^{100} f(x) d x\)?

The graph shows the velocity \((in\ m/s)\) of an electric autonomous vehicle moving along a straight track. At \(t=0\) the vehicle is at the charging station. (a) How far is the vehicle from the charging station when \(t=2,4,6,8,10\), and \(12\) ? (b) At what times is the vehicle farthest from the charging station? (c) What is the total distance traveled by the vehicle? Equation Transcription: Text Transcription: (in m/s) t=0 t=2,4,6,8,10 12

The velocity function (in mys) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. \(v(t)=3 t-5, \quad 0 \leqslant t \leqslant 3\)

The velocity function (in m/s) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. \(v(t)=t^{2}-2 t-3, \quad 2 \leqslant t \leqslant 4\)

The acceleration function (in \(\mathrm{m} / \mathrm{s}^{2}\) ) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time \(t\) and (b) the distance traveled during the given time interval. \(a(t)=t+4, \quad v(0)=5, \quad 0 \leqslant t \leqslant 10\) Equation Transcription: ?? Text Transcription: (in m/s2) t a(t)=t+4, v(0)=5, 0 less than or equal slant t less than or equal slant 10

The acceleration function (in \(\mathrm{m} / \mathrm{s}^{2}\) ) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time \(t\) and (b) the distance traveled during the given time interval. \(a(t)=2 t+3, \quad v(0)=-4, \quad 0 \leqslant t \leqslant 3\) Equation Transcription: ?? Text Transcription: (in m/s2) t a(t)=2t+3, v(0)=-4, 0 less than or equal slant t less than or equal slant 3

he linear density of a rod of length \(4 \mathrm{~m}\) is given by \(\rho(x)=9+2 \sqrt{x}\) measured in kilograms per meter, where \(x\) is measured in meters from one end of the rod. Find the total mass of the rod. Equation Transcription: Text Transcription: x rho (x)=9+2 square root x

Water flows from the bottom of a storage tank at a rate of \(r(t)=200-4 t\) liters per minute, where \(0 \leqslant t \leqslant 50\). Find the amount of water that flows from the tank during the first 10 minutes. ________________ Equation Transcription: ?? Text Transcription: r(t)=200-4t 0 less than or equal slant t less than or equal slant 50

The velocity of a car was read from its speedometer at \(10-second\) intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. Equation Transcription: Text Transcription: 10-second

Suppose that a volcano is erupting and readings of the rate \(r(t)\) at which solid materials are spewed into the atmosphere are given in the table. The time \(t\) is measured in seconds and the units for \(r(t)\) are tonnes (metric tons) per second. (a) Give upper and lower estimates for the total quantity \(Q(6)\) of erupted materials after six seconds. (b) Use the Midpoint Rule to estimate \(Q(6)\). Equation Transcription: Text Transcription: r(t) t Q(6)

The marginal cost of manufacturing \(x\) yards of a certain fabric is \(C^{\prime}(x)=3-0.01 x+0.000006 x^{2}\) (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards. ________________ Equation Transcription: Text Transcription: x C'(x) = 3 ? 0.01x + 0.000006 x^2

Water flows into and out of a storage tank. A graph of the rate of change r(t) of the volume of water in the tank, in liters per day, is shown. If the amount of water in the tank at time t = 0 is 25,000 L, use the Midpoint Rule to estimate the amount of water in the tank four days later.

The graph of a car's acceleration \(a(t)\), measured in \(\mathrm{ft} / \mathrm{s}^{2}\), is shown. Use the Midpoint Rule to estimate the increase in the velocity of the car during the six-second time interval. Equation Transcription: Text Transcription: a(t) ft/s^2

Lake Lanier in Georgia, USA, is a reservoir created by Buford Dam on the Chattahoochee River. The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at \(7: 30 \mathrm{AM}\) by the US Army Corps of Engineers. Use the Midpoint Rule to estimate the amount of water that flowed into Lake Lanier from July 18 th, 2013 , at \(7: 30 \mathrm{AM}\) to July 26 th at \(7: 30 \mathrm{AM}\). Equation Transcription: Text Transcription: 7:30 am

A bacteria population is 4000 at time \(t=0\) and its rate of growth is \(1000 \cdot 2^{t}\) bacteria per hour after \(t\) hours. What is the population after one hour? Equation Transcription: Text Transcription: t=0 1000 dot 2^t t

Shown is the graph of traffic on an Internet service provider’s T1 data line from midnight to 8:00 am. D is the data throughput, measured in megabits per second. Use the Midpoint Rule to estimate the total amount of data transmitted during that time period.

Shown is a graph of the electric power consumption in the New England states (Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont) for October 22, 2010 (???? is measured in gigawatts and ???? is measured in hours starting at midnight). Use the fact that power is the rate of change of energy to estimate the electric energy used on that day.

In 1992 the space shuttle Endeavour was launched on mission STS-49 in order to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to model these data by a third-degree polynomial. (b) Use the model in part (a) to estimate the height reached by the Endeavour, 125 seconds after liftoff.