Question 1

Evaluate the integral.
$$\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$$

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 2 \pi } { 3 }$ 
A)  $\frac { \pi } { 4 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 2 \pi } { 3 }$

Question 2

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 $\int _ { 0 } ^ { 2 } \left( 2 y - y ^ { 2 } \right) d y$.
Find the area..
 
 Select the correct answer.

Accepted Answer

A)  $\frac { 175 } { 6 }$ 
B)  4 
C)  9 
D)  $\frac { 4 } { 3 }$ 
E)  6 
A)  $\frac { 175 } { 6 }$ 
B)  4 
C)  9 
D)  $\frac { 4 } { 3 }$ 
E)  6

Question 3

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$.
$f ( x ) = x ^ { 2 } , \quad [ 0,6 ] , \quad c _ { k }$ is the left endpoint

Accepted Answer

The answer of Use the definition of area to find...

Question 4

Find the indefinite integral. $$\int \frac { 1 - \cos ^ { 2 } x } { \sin x } d x$$ Select the correct answer.

Accepted Answer

A)  $\sin x + C$ 
B)  $2 x + C$ 
C)  $- \csc x + C$ 
D)  $- \cos x + C$ 
A)  $\sin x + C$ 
B)  $2 x + C$ 
C)  $- \csc x + C$ 
D)  $- \cos x + C$

Question 5

Find the integral.
$$\int x ^ { 4 } \ln x d x$$

Accepted Answer

The answer of Find the integral.
\[\int x ^ { 4...

Question 6

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find a lower estimate for the distance that she traveled during these three seconds.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline t(\mathrm{~s}) & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \
\hline(\mathrm{ft} / \mathrm{s}) & 0 & 2.1 & 5 & 6.8 & 9.7 & 12.9 & 17.5 \
\hline
\end{array}$

Accepted Answer

The answer of The speed of a runner increased steadily...

Question 7

Approximate the area under the curve $y = \sin x$ from 0 to $\frac { \pi } { 2 }$ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Accepted Answer

A)  $3.09$ 
B)  $1.09$ 
C)  $4.09$ 
D)  $0.09$ 
E)  $2.09$ 
A)  $3.09$ 
B)  $1.09$ 
C)  $4.09$ 
D)  $0.09$ 
E)  $2.09$

Question 8

Find the integral.
$$\int \cos ^ { 3 } x \sin ^ { 2 } x d x$$

Accepted Answer

The answer of Find the integral.
\[\int \cos ^ { 3...

Question 9

A table of values of an increasing function $f ( x )$ is shown. Use the table to find an upper estimate of:
$$\int _ { 0 } ^ { 25 } f ( x ) d x$$
$\begin{array}{|c|c|}
\hline x & f(x) \
\hline 0 & -46 \
\hline 5 & -30 \
\hline 10 & -20 \
\hline 15 & 4 \
\hline 20 & 15 \
\hline 25 & 20 \
\hline
\end{array}$

Accepted Answer

The answer of A table of values of an increasing...

Question 10

Use the given graph of $f$ to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

Accepted Answer

The answer of Use the given graph of $f$ to...

Question 11

$$\text { Find the derivative of the function } \int _ { x } ^ { x ^ { 6 } } \ln t d t \text {. }$$

Accepted Answer

The answer of \[\text { Find the derivative of the...

Question 12

Find the indefinite integral $\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x$.

Accepted Answer

A)  $\frac { x ^ { 3 } - 25 x } { x ^ { 3 } + 25 x } + C$ 
B) 
$x - 10 \arctan \left( \frac { x } { 5 } \right) + C$ 
C) 
$x - 6 \arctan \left( \frac { x } { 5 } \right) + C$ 
D)  $\frac { x ^ { 3 } - 75 x } { x ^ { 3 } + 75 x } + C$ 
A)  $\frac { x ^ { 3 } - 25 x } { x ^ { 3 } + 25 x } + C$ 
B) 
$x - 10 \arctan \left( \frac { x } { 5 } \right) + C$ 
C) 
$x - 6 \arctan \left( \frac { x } { 5 } \right) + C$ 
D)  $\frac { x ^ { 3 } - 75 x } { x ^ { 3 } + 75 x } + C$

Question 13

Find the derivative of the function.
$$g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$$

Accepted Answer

The answer of Find the derivative of the function.
\[g (...

Question 14

Find the area of the region under the graph of $f$ on $[ a , b ]$.
$$f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]$$

Accepted Answer

A)  $- 3$ 
B)  $- 6$ 
C)  6 
D)  3 
A)  $- 3$ 
B)  $- 6$ 
C)  6 
D)  3

Question 15

$$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$$

Accepted Answer

@#LAT-DLM&
@#LAT-DLM& Area of

Question 16

Use the following property of the definite integral to estimate the definite integral $\int _ { z / 6 } ^ { \pi / 4 } 6 \sin x d x$ :
If $m \leq f ( x ) \leq M$ on $[ a , b ]$, then
$$m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$$

Accepted Answer

A) 
$\frac { \pi } { 2 } \leq \int _ { s / 6 } ^ { x / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 2 }$ 
B)  $\frac { \sqrt { 2 } \pi } { 4 } \leq \int _ { z / 6 } ^ { s / 4 } 6 \sin x d x \leq \frac { \pi } { 4 }$ 
C) 
$\frac { \pi } { 4 } \leq \int _ { s / 6 } ^ { s / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 4 }$ 
D)  $\frac { \sqrt { 2 } \pi } { 2 } \leq \int _ { s / 6 } ^ { \pi / 4 } 6 \sin x d x \leq \frac { \pi } { 2 }$ 
A) 
$\frac { \pi } { 2 } \leq \int _ { s / 6 } ^ { x / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 2 }$ 
B)  $\frac { \sqrt { 2 } \pi } { 4 } \leq \int _ { z / 6 } ^ { s / 4 } 6 \sin x d x \leq \frac { \pi } { 4 }$ 
C) 
$\frac { \pi } { 4 } \leq \int _ { s / 6 } ^ { s / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 4 }$ 
D)  $\frac { \sqrt { 2 } \pi } { 2 } \leq \int _ { s / 6 } ^ { \pi / 4 } 6 \sin x d x \leq \frac { \pi } { 2 }$

Question 17

Evaluate the integral.
$$\int _ { 0 } ^ { \frac { 1 } { 3 } } \frac { d r } { \sqrt { 1 - r ^ { 2 } } }$$

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { 0 }...

Question 18

Find the general indefinite integral.
$$\int \frac { \sin 18 t } { \sin 9 t } d t$$

Accepted Answer

The answer of Find the general indefinite integral.
\[\int \frac {...

Question 19

Evaluate the integral.
$\int _ { x / 3 } ^ { x / 7 } \sin t d t$

Accepted Answer

A)  $- 0.500$ 
B)  $- 1.000$ 
C)  $- 0.400$ 
D)  $0.250$ 
E)  $1.000$ 
A)  $- 0.500$ 
B)  $- 1.000$ 
C)  $- 0.400$ 
D)  $0.250$ 
E)  $1.000$

Question 20

Find the indefinite integral.
$$\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$$

Accepted Answer

A)  $2 \left( x ^ { 1 / 3 } - 2 \right) ^ { 3 / 2 } + C$ 
B)  $2 x ^ { 1 / 3 } \left( x ^ { 1 / 3 } - 2 \right) ^ { 3 / 2 } + C$ 
C)  $\frac { 1 } { 2 } \left( x ^ { 1 / 3 } - 2 \right) ^ { 3 / 2 } + C$ 
D)  $x ^ { 1 / 3 } \left( x ^ { 1 / 3 } - 2 \right) ^ { 3 / 2 } + C$ 
A)  $2 \left( x ^ { 1 / 3 } - 2 \right) ^ { 3 / 2 } + C$ 
B)  $2 x ^ { 1 / 3 } \left( x ^ { 1 / 3 } - 2 \right) ^ { 3 / 2 } + C$ 
C)  $\frac { 1 } { 2 } \left( x ^ { 1 / 3 } - 2 \right) ^ { 3 / 2 } + C$ 
D)  $x ^ { 1 / 3 } \left( x ^ { 1 / 3 } - 2 \right) ^ { 3 / 2 } + C$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$ . $f ( x ) = x ^ { 2 } , \quad [ 0,6 ] , \quad c _ { k }$ is the left endpoint

Find the indefinite integral. $\int \frac { 1 - \cos ^ { 2 } x } { \sin x } d x$ Select the correct answer.

Find the integral. $\int x ^ { 4 } \ln x d x$

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find a lower estimate for the distance that she traveled during these three seconds. t() 0 0.5 1.0 1.5 2.0 2.5 3.0 (/) 0 2.1 5 6.8 9.7 12.9 17.5

Approximate the area under the curve $y = \sin x$ from 0 to $\frac { \pi } { 2 }$ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Find the integral. $\int \cos ^ { 3 } x \sin ^ { 2 } x d x$

A table of values of an increasing function $f ( x )$ is shown. Use the table to find an upper estimate of: $\int _ { 0 } ^ { 25 } f ( x ) d x$ x f(x) 0 -46 5 -30 10 -20 15 4 20 15 25 20

Use the given graph of $f$ to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

$\text { Find the derivative of the function } \int _ { x } ^ { x ^ { 6 } } \ln t d t \text {. }$

Find the indefinite integral $\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x$ .

Find the derivative of the function. $g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]$

$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$

Use the following property of the definite integral to estimate the definite integral $\int _ { z / 6 } ^ { \pi / 4 } 6 \sin x d x$ : If $m \leq f ( x ) \leq M$ on $[ a , b ]$ , then $m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$

Evaluate the integral. $\int _ { 0 } ^ { \frac { 1 } { 3 } } \frac { d r } { \sqrt { 1 - r ^ { 2 } } }$

Find the general indefinite integral. $\int \frac { \sin 18 t } { \sin 9 t } d t$

Evaluate the integral. $\int _ { x / 3 } ^ { x / 7 } \sin t d t$

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Exam 4: Applications of Differentiation

Evaluate the integral. ∫01/24dr1−r2\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }∫01/2​1−r2​4dr​

Use the definition of area to find the area of the region under the graph of fff on [a,b][ a , b ][a,b] using the indicated choice of ckc _ { k }ck​ . f(x)=x2,[0,6],ckf ( x ) = x ^ { 2 } , \quad [ 0,6 ] , \quad c _ { k }f(x)=x2,[0,6],ck​ is the left endpoint

Find the indefinite integral. ∫1−cos⁡2xsin⁡xdx\int \frac { 1 - \cos ^ { 2 } x } { \sin x } d x∫sinx1−cos2x​dx Select the correct answer.

Find the integral. ∫x4ln⁡xdx\int x ^ { 4 } \ln x d x∫x4lnxdx

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find a lower estimate for the distance that she traveled during these three seconds. t() 0 0.5 1.0 1.5 2.0 2.5 3.0 (/) 0 2.1 5 6.8 9.7 12.9 17.5

Approximate the area under the curve y=sin⁡xy = \sin xy=sinx from 0 to π2\frac { \pi } { 2 }2π​ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Find the integral. ∫cos⁡3xsin⁡2xdx\int \cos ^ { 3 } x \sin ^ { 2 } x d x∫cos3xsin2xdx

A table of values of an increasing function f(x)f ( x )f(x) is shown. Use the table to find an upper estimate of: ∫025f(x)dx\int _ { 0 } ^ { 25 } f ( x ) d x∫025​f(x)dx x f(x) 0 -46 5 -30 10 -20 15 4 20 15 25 20

Use the given graph of fff to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

Find the derivative of the function ∫xx6ln⁡tdt. \text { Find the derivative of the function } \int _ { x } ^ { x ^ { 6 } } \ln t d t \text {. } Find the derivative of the function ∫xx6​lntdt.

Find the indefinite integral ∫x2−25x2+25dx\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x∫x2+25x2−25​dx .

Find the derivative of the function. g(x)=∫4x241+t2dtg ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d tg(x)=∫4x2​41+t2​dt

Find the area of the region under the graph of fff on [a,b][ a , b ][a,b] . f(x)=x2−2x+2;[−1,2]f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]f(x)=x2−2x+2;[−1,2]

Evaluate ∫03(x+9−x2)dx by interpreting it in terms of areas. \text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. } Evaluate ∫03​(x+9−x2​)dx by interpreting it in terms of areas.

Evaluate the integral. ∫013dr1−r2\int _ { 0 } ^ { \frac { 1 } { 3 } } \frac { d r } { \sqrt { 1 - r ^ { 2 } } }∫031​​1−r2​dr​

Find the general indefinite integral. ∫sin⁡18tsin⁡9tdt\int \frac { \sin 18 t } { \sin 9 t } d t∫sin9tsin18t​dt

Evaluate the integral. ∫x/3x/7sin⁡tdt\int _ { x / 3 } ^ { x / 7 } \sin t d t∫x/3x/7​sintdt

Find the indefinite integral. ∫x−2/3x1/3−2dx\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x∫x−2/3x1/3−2​dx

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$ . $f ( x ) = x ^ { 2 } , \quad [ 0,6 ] , \quad c _ { k }$ is the left endpoint

Find the indefinite integral. $\int \frac { 1 - \cos ^ { 2 } x } { \sin x } d x$ Select the correct answer.

Find the integral. $\int x ^ { 4 } \ln x d x$

Approximate the area under the curve $y = \sin x$ from 0 to $\frac { \pi } { 2 }$ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Find the integral. $\int \cos ^ { 3 } x \sin ^ { 2 } x d x$

A table of values of an increasing function $f ( x )$ is shown. Use the table to find an upper estimate of: $\int _ { 0 } ^ { 25 } f ( x ) d x$ x f(x) 0 -46 5 -30 10 -20 15 4 20 15 25 20

Use the given graph of $f$ to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

$\text { Find the derivative of the function } \int _ { x } ^ { x ^ { 6 } } \ln t d t \text {. }$

Find the indefinite integral $\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x$ .

Find the derivative of the function. $g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]$

$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$

Evaluate the integral. $\int _ { 0 } ^ { \frac { 1 } { 3 } } \frac { d r } { \sqrt { 1 - r ^ { 2 } } }$

Find the general indefinite integral. $\int \frac { \sin 18 t } { \sin 9 t } d t$

Evaluate the integral. $\int _ { x / 3 } ^ { x / 7 } \sin t d t$

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$