Question 1

Find the area of the shaded region. Round your answer to one decimal place.
​

Accepted Answer

The answer of Find the area of the shaded region....

Question 2

Find the area of the shaded region.
​
​

Accepted Answer

The answer of Find the area of the shaded region.
​
​...

Question 3

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
​
The area of the region bounded by the graphs of  and  and the vertical lines  and  is given by  .

Accepted Answer

Since f and g are continuous functions and 11eb651f_cb13_dc04_b0a1_b5ffe389b35a_TB8255_11 on the interval [0, 2], then the area of the region bounded above by 11eb651f_cb13_dc05_b0a1_092d55a89eb8_TB8255_11 and below by 11eb651f_cb13_dc06_b0a1_df37d0869a18_TB8255_11 on [0, 2] is given by 11eb651f_cb13_dc07_b0a1_879cf4d28df8_TB8255_11 .

Question 4

In tests conducted by Auto Test Magazine on two identical models of the Phoenix Elite - one equipped with a standard engine and the other with a turbo-charger - it was found that the acceleration of the former is given by
​  ​
ft/sec, t seconds after starting from rest at full throttle, whereas the acceleration of the latter is given by
​
​  ​
ft/sec. How much faster is the turbo-charged model moving than the model with the standard engine at the end of a 10-second test run at full throttle?
​
__________ ft/sec

Accepted Answer

The answer of In tests conducted by Auto Test Magazine...

Question 5

In an endeavor to curb population growth in a certain country, the government has decided to launch an extensive propaganda campaign. Without curbs, the government expects the rate of population growth to have been
​  ​
thousand people per year t years from now, over the next 5 years. However, successful implementation of the proposed campaign is expected to result in a population growth rate of
​  ​
thousand people per year t years from now, over the next 5 years. Assuming that the campaign is mounted, how many fewer people will there be in that country 5 years from now then there would have been if no curbs had been imposed?
​
Please round your answer to the nearest whole number.
​
__________ thousand people

Accepted Answer

The answer of In an endeavor to curb population growth...

Question 6

Find the area of the shaded region.
​
​

Accepted Answer

The answer of Find the area of the shaded region.
​
​...

Question 7

Because of the increasingly important role played by coal as a viable alternative energy source, the production of coal has been growing at the rate of
​  ​
billion metric tons/year t years from 1960 (which corresponds to  ). Had it not been for the energy crisis, the rate of production of coal since 1960 might have been only
​  ​
billion metric tons/year t years from 1960. Determine how much additional coal was produced between 1960 and the end of the century as an alternate energy source. Round your answer to one decimal place.
​
__________ billion metric tons

Accepted Answer

The answer of Because of the increasingly important role played...

Question 8

Evaluate the given definite integral. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Evaluate the given definite integral. ​ ...

Question 9

Evaluate the given definite integral. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

Using the substitution $x=3\sin^2	heta$, we have $dx=6\sin	heta\cos	heta d	heta$ and the limits of integration become $	heta=0$ to $	heta=\frac{\pi}{2}$. Thus, the integral becomes 
\begin{align*}
\int_0^3x\sqrt{3-x}\,dx &= \int_0^{\frac{\pi}{2}} 3\sin^2	heta\sqrt{3-3\sin^2	heta}\cdot 6\sin	heta\cos	heta\,d	heta\
&=54\int_0^{\frac{\pi}{2}} \sin^3	heta\cos^2	heta\,d	heta\
&=54\int_0^{\frac{\pi}{2}} \sin^3	heta(1-\sin^2	heta)\,d	heta\
&=54\int_0^1 u^3(1-u^2)\frac{du}{2\sqrt{u}}\
&=\frac{27}{2}\int_0^1 u^\frac{5}{2}-u^\frac{7}{2}\,du\
&=\frac{27}{2}\left[\frac{2}{7}u^\frac{7}{2}-\frac{2}{9}u^\frac{9}{2}ight]_0^1\
&=\frac{27}{2}\left(\frac{2}{7}\cdot 1^\frac{7}{2}-\frac{2}{9}\cdot 1^\frac{9}{2}ight)\
&=\frac{27}{7}.
\end{align*}

Question 10

Evaluate the given definite integral. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Evaluate the given definite integral. ​ ...

Question 11

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
​
The area of the region bounded by the graphs of  and  and the vertical lines  and  is given by  .
​

Accepted Answer

The answer of Determine whether the statement is true or...

Question 12

Evaluate the definite integral. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Evaluate the definite integral. ​  ​
A)...

Question 13

Evaluate the given definite integral. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

Using the power rule of integration, we get 
$$\int_1^4 (2x+1)^2 dx = \frac{(2x+1)^3}{6}\bigg|_1^4 = \frac{(9^3-3^3)}{6} = \frac{522}{3}.$$ Simplifying, we get $\frac{174}{1} = 174.$ Therefore, the correct option is C.

Question 14

Evaluate the given definite integral. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Evaluate the given definite integral. ​ ...

Question 15

Find the area of the shaded region.
​

Accepted Answer

The answer of Find the area of the shaded region.
​...

Question 16

Evaluate the definite integral. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Evaluate the definite integral. ​  ​
A)...

Question 17

Evaluate the given definite integral. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

Using the power rule of integration, we have

∫(1/x^3) dx = (-1/2)x^-2 + C

Evaluating this definite integral from 2 to 4, we get:

(-1/2)(4)^-2 - (-1/2)(2)^-2 = 1/16 - 1/4 = -3/16

Therefore, the correct answer is (A).

Question 18

Evaluate the definite integral. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The integral can be rewritten as:
$$\int_0^{\pi} \sqrt{\frac{1}{2}(1-\cos x)} dx$$
Using the identity $1-\cos 2x = 2\sin^2 x$, we can simplify the integrand:
$$\int_0^{\pi} \sqrt{\frac{1}{2}(1-\cos x)} dx = \int_0^{\pi} \sqrt{\frac{1}{2}\sin^2 x} dx = \int_0^{\pi} \frac{\sin x}{\sqrt{2}} dx$$
Then, using the substitution $u=\cos x$, $du=-\sin x dx$, we get:
$$\int_0^{\pi} \frac{\sin x}{\sqrt{2}} dx = -\sqrt{2}\int_1^{-1} \frac{du}{\sqrt{1-u^2}} = \sqrt{2} \int_{-1}^1 \frac{du}{\sqrt{1-u^2}}$$
This is the integral for the arc length of the half-unit circle, which evaluates to $\pi\sqrt{2}$, so the answer is (B).

Question 19

Evaluate the given definite integral. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Evaluate the given definite integral. ​ ...

Question 20

Evaluate the given definite integral. ​  ​
A) 
B) 
C) 
D)

Accepted Answer

The answer of Evaluate the given definite integral. ​ ...

Quiz 6: Integration

Find the area of the shaded region. Round your answer to one decimal place.

Find the area of the shaded region.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of the region bounded by the graphs of and and the vertical lines and is given by .

Find the area of the shaded region.

Evaluate the given definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of the region bounded by the graphs of and and the vertical lines and is given by .

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Find the area of the shaded region.

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Quiz 6: Integration

Find the area of the shaded region. Round your answer to one decimal place. ​

Find the area of the shaded region. ​ ​

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. ​ The area of the region bounded by the graphs of and and the vertical lines and is given by .

Find the area of the shaded region. ​ ​

Evaluate the given definite integral. ​ ​

Evaluate the given definite integral. ​ ​

Evaluate the given definite integral. ​ ​

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. ​ The area of the region bounded by the graphs of and and the vertical lines and is given by . ​

Evaluate the definite integral. ​ ​

Evaluate the given definite integral. ​ ​

Evaluate the given definite integral. ​ ​

Find the area of the shaded region. ​

Evaluate the definite integral. ​ ​

Evaluate the given definite integral. ​ ​

Evaluate the definite integral. ​ ​

Evaluate the given definite integral. ​ ​

Evaluate the given definite integral. ​ ​

Find the area of the shaded region. Round your answer to one decimal place.

Find the area of the shaded region.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of the region bounded by the graphs of and and the vertical lines and is given by .

Find the area of the shaded region.

Evaluate the given definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of the region bounded by the graphs of and and the vertical lines and is given by .

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Find the area of the shaded region.

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.