Question 1

Determine the area of the region bounded by the curve $y = x ^ { 3 } - 3 x ^ { 2 } + x + 5$ and the line y = x + 5.

Accepted Answer

The answer of Determine the area of the region bounded...

Question 2

Evaluate $\int \frac { 2 x - 3 } { x ^ { 4 } } d x$ .

Accepted Answer

The answer of Evaluate \(\int \frac { 2 x -...

Question 3

$\int ( x + 6 ) ^ { 10 } d x = \frac { ( x + 6 ) ^ { 11 } } { 11 } + C$

Accepted Answer

A) True 
 B)False

Question 4

If $\int _ { a } ^ { b } f ( x ) d x = - 7$ and $\int _ { a } ^ { b } g ( x ) d x = 3$ , find $\int _ { b } ^ { a } [ f ( x ) - g ( x ) ] d x$ .

Accepted Answer

A)  10 
B)  -21 
C)  -10 
D)  -4 
A)  10 
B)  -21 
C)  -10 
D)  -4

Question 5

The velocity v(t) (t)= x'(t) at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. $x ^ { t } ( t ) = - 4 ( 3 t + 1 ) ^ { 1 / 2 }$ ; x(0) = 4
Find: (a) The position x(t) at time t.
(b) The position of the object at time t = 6.
(c) The time when the object is at x = 1.Round answers for parts (b) and (c) to one decimal place.

Accepted Answer

(a) @#LAT-DLM& (b) x

Question 6

Solve the given initial value problem for y = f (x). $$\frac { d y } { d x } = \frac { x - 5 } { x ^ { 2 } - 10 x + 8 }$$ where y = 6 when x = 2

Accepted Answer

A)  $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right|$ 
B)  $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| + 6$ 
C)  $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| - \frac { 1 } { 2 } \ln 8 + 6$ 
D)  $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| - \frac { 1 } { 2 } \ln 8$ 
A)  $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right|$ 
B)  $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| + 6$ 
C)  $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| - \frac { 1 } { 2 } \ln 8 + 6$ 
D)  $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| - \frac { 1 } { 2 } \ln 8$

Question 7

Specify the substitution you would choose to evaluate the integrals. $$\int ( t + 4 ) e ^ { t ^ { 2 } + 8 t + 1 } d t$$

Accepted Answer

A)  $u = t + 4$ 
B)  $U = e ^ { t ^ { 2 } + 8 t + 1 }$ 
C)  $u = t ^ { 2 } + 8 t + 1$ 
D)  $u = ( t + 4 ) e$ 
A)  $u = t + 4$ 
B)  $U = e ^ { t ^ { 2 } + 8 t + 1 }$ 
C)  $u = t ^ { 2 } + 8 t + 1$ 
D)  $u = ( t + 4 ) e$

Question 8

A study indicates that x months from now the population of a certain city will be increasing at the rate of $( 4 + 3 x ) x ^ { - 1 / 2 }$ people per month. By how much will the population increase over the next 4 months?

Accepted Answer

The answer of A study indicates that x months from...

Question 9

$\int ( x - 6 ) ^ { 7 } d x = \frac { ( x - 6 ) ^ { 8 } } { 8 } + C$

Accepted Answer

A) True 
 B)False

Question 10

Evaluate $\int \left( 5 x ^ { 2 } - e ^ { x } \right) d x$ .

Accepted Answer

The answer of Evaluate \(\int \left( 5 x ^ {...

Question 11

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the curve $y = \sqrt { 16 - x ^ { 2 } }$ from x = -4 to x = 4.

Accepted Answer

A)  $16 \pi$ 
B)  $\frac { 64 } { 3 } \pi$ 
C)  $\frac { 2.56 } { 3 } \pi$ 
D)  $64 \pi$ 
A)  $16 \pi$ 
B)  $\frac { 64 } { 3 } \pi$ 
C)  $\frac { 2.56 } { 3 } \pi$ 
D)  $64 \pi$

Question 12

Evaluate $\int _ { - 1 } ^ { 1 } x ^ { 15 } d x$ .

Accepted Answer

The answer of Evaluate \(\int _ { - 1 }...

Question 13

Find the average value of the function $f ( x ) = \frac { 2 x } { 2 x + 3 }$ over the interval [2, 6]. Round your answer to two decimal places.

Accepted Answer

A)  0.71 
B)  2.86 
C)  1.29 
D)  1.71 
A)  0.71 
B)  2.86 
C)  1.29 
D)  1.71

Question 14

Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. $L ( x ) = 0.7 x ^ { 2 } + 0.3 x$

Accepted Answer

A)  1.7667 
B)  0.2333 
C)  0.2000 
D)  0.2800 
A)  1.7667 
B)  0.2333 
C)  0.2000 
D)  0.2800

Question 15

The velocity $$v ( t ) ( \mathrm { t } ) = x ^ { \prime } ( t )$$ at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. $$x ^ { \prime } ( t ) = - 2 ( 5 t + 1 ) ^ { 1 / 2 }$$ ; x(0) = 3 
Find: 
(a) The position x(t) at time t. 
(b) The position of the object at time t = 6. 
(c) The time when the object is at x = 2. Round answers for parts (b) and (c) to one decimal place.

Accepted Answer

A)  (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 } + \frac { 49 } { 15 }$ 
(b) x(6) = -42.8
(c) t = 0.4 
B)  (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 }$ 
(b) x(6) = -46.0
(c) t = -0.2 
C)  (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 } + \frac { 49 } { 15 }$ 
(b) x(6) = 0.7
(c) t = -6.5 
D)  (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 }$ 
(b) x(6) = 1.4
(c) t = -9.7 
A)  (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 } + \frac { 49 } { 15 }$ 
(b) x(6) = -42.8
(c) t = 0.4 
B)  (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 }$ 
(b) x(6) = -46.0
(c) t = -0.2 
C)  (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 } + \frac { 49 } { 15 }$ 
(b) x(6) = 0.7
(c) t = -6.5 
D)  (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 }$ 
(b) x(6) = 1.4
(c) t = -9.7

Question 16

Evaluate $\int _ { - 2 } ^ { 2 } \left( x ^ { 5 } + 3 x ^ { 3 } - 2 x \right) d x$ .

Accepted Answer

The answer of Evaluate \(\int _ { - 2 }...

Question 17

The velocity $$v ( t ) ( \mathrm { t } ) = x ^ { \prime } ( t )$$ at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. $x ^ { t } ( t ) = - 3 ( 3 t + 1 ) ^ { 1 / 2 }$ ; x(0) = 4
Find:
(a) The position x(t) at time t.
(b) The position of the object at time t = 1.
(c) The time when the object is at x = 2.Round answers for parts (b) and (c) to one decimal place.

Accepted Answer

A)  (a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 }$ 
(b) x(1) = -5.3
(c) t = -0.3 
B)  (a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 } + \frac { 14 } { 3 }$ 
(b) x(1) = 0.7
(c) t = -7.7 
C)  (a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 }$ 
(b) x(1) = 0.1
(c) t = -12.3 
D)  (a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 } + \frac { 14 } { 3 }$ 
(b) x(1) = -0.7
(c) t = 0.5 
A)  (a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 }$ 
(b) x(1) = -5.3
(c) t = -0.3 
B)  (a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 } + \frac { 14 } { 3 }$ 
(b) x(1) = 0.7
(c) t = -7.7 
C)  (a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 }$ 
(b) x(1) = 0.1
(c) t = -12.3 
D)  (a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 } + \frac { 14 } { 3 }$ 
(b) x(1) = -0.7
(c) t = 0.5

Question 18

Evaluate $\int x ^ { 2 } e ^ { x ^ { 3 } } d x$ .

Accepted Answer

The answer of Evaluate \(\int x ^ { 2 }...

Question 19

Determine the area of the region bounded by the line y = x and the curve $y = x ^ { 3 }$ .

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 5 } { 8 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 7 } { 8 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 5 } { 8 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 7 } { 8 }$

Question 20

Evaluate $\int _ { 2 } ^ { 3 } \frac { 2 x - 3 } { x ^ { 4 } } d x$ .

Accepted Answer

The answer of Evaluate \(\int _ { 2 } ^...

Determine the area of the region bounded by the curve $y = x ^ { 3 } - 3 x ^ { 2 } + x + 5$ and the line y = x + 5.

Evaluate $\int \frac { 2 x - 3 } { x ^ { 4 } } d x$ .

$\int ( x + 6 ) ^ { 10 } d x = \frac { ( x + 6 ) ^ { 11 } } { 11 } + C$

If $\int _ { a } ^ { b } f ( x ) d x = - 7$ and $\int _ { a } ^ { b } g ( x ) d x = 3$ , find $\int _ { b } ^ { a } [ f ( x ) - g ( x ) ] d x$ .

Solve the given initial value problem for y = f (x). $\frac { d y } { d x } = \frac { x - 5 } { x ^ { 2 } - 10 x + 8 }$ where y = 6 when x = 2

Specify the substitution you would choose to evaluate the integrals. $\int ( t + 4 ) e ^ { t ^ { 2 } + 8 t + 1 } d t$

A study indicates that x months from now the population of a certain city will be increasing at the rate of $( 4 + 3 x ) x ^ { - 1 / 2 }$ people per month. By how much will the population increase over the next 4 months?

$\int ( x - 6 ) ^ { 7 } d x = \frac { ( x - 6 ) ^ { 8 } } { 8 } + C$

Evaluate $\int \left( 5 x ^ { 2 } - e ^ { x } \right) d x$ .

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the curve $y = \sqrt { 16 - x ^ { 2 } }$ from x = -4 to x = 4.

Evaluate $\int _ { - 1 } ^ { 1 } x ^ { 15 } d x$ .

Find the average value of the function $f ( x ) = \frac { 2 x } { 2 x + 3 }$ over the interval [2, 6]. Round your answer to two decimal places.

Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. $L ( x ) = 0.7 x ^ { 2 } + 0.3 x$

Evaluate $\int _ { - 2 } ^ { 2 } \left( x ^ { 5 } + 3 x ^ { 3 } - 2 x \right) d x$ .

Evaluate $\int x ^ { 2 } e ^ { x ^ { 3 } } d x$ .

Determine the area of the region bounded by the line y = x and the curve $y = x ^ { 3 }$ .

Evaluate $\int _ { 2 } ^ { 3 } \frac { 2 x - 3 } { x ^ { 4 } } d x$ .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Exam 5: Integration

Determine the area of the region bounded by the curve y=x3−3x2+x+5y = x ^ { 3 } - 3 x ^ { 2 } + x + 5y=x3−3x2+x+5 and the line y = x + 5.

Evaluate ∫2x−3x4dx\int \frac { 2 x - 3 } { x ^ { 4 } } d x∫x42x−3​dx .

∫(x+6)10dx=(x+6)1111+C\int ( x + 6 ) ^ { 10 } d x = \frac { ( x + 6 ) ^ { 11 } } { 11 } + C∫(x+6)10dx=11(x+6)11​+C

If ∫abf(x)dx=−7\int _ { a } ^ { b } f ( x ) d x = - 7∫ab​f(x)dx=−7 and ∫abg(x)dx=3\int _ { a } ^ { b } g ( x ) d x = 3∫ab​g(x)dx=3 , find ∫ba[f(x)−g(x)]dx\int _ { b } ^ { a } [ f ( x ) - g ( x ) ] d x∫ba​[f(x)−g(x)]dx .

Solve the given initial value problem for y = f (x). dydx=x−5x2−10x+8\frac { d y } { d x } = \frac { x - 5 } { x ^ { 2 } - 10 x + 8 }dxdy​=x2−10x+8x−5​ where y = 6 when x = 2

Specify the substitution you would choose to evaluate the integrals. ∫(t+4)et2+8t+1dt\int ( t + 4 ) e ^ { t ^ { 2 } + 8 t + 1 } d t∫(t+4)et2+8t+1dt

A study indicates that x months from now the population of a certain city will be increasing at the rate of (4+3x)x−1/2( 4 + 3 x ) x ^ { - 1 / 2 }(4+3x)x−1/2 people per month. By how much will the population increase over the next 4 months?

∫(x−6)7dx=(x−6)88+C\int ( x - 6 ) ^ { 7 } d x = \frac { ( x - 6 ) ^ { 8 } } { 8 } + C∫(x−6)7dx=8(x−6)8​+C

Evaluate ∫(5x2−ex)dx\int \left( 5 x ^ { 2 } - e ^ { x } \right) d x∫(5x2−ex)dx .

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the curve y=16−x2y = \sqrt { 16 - x ^ { 2 } }y=16−x2​ from x = -4 to x = 4.

Evaluate ∫−11x15dx\int _ { - 1 } ^ { 1 } x ^ { 15 } d x∫−11​x15dx .

Find the average value of the function f(x)=2x2x+3f ( x ) = \frac { 2 x } { 2 x + 3 }f(x)=2x+32x​ over the interval [2, 6]. Round your answer to two decimal places.

Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. L(x)=0.7x2+0.3xL ( x ) = 0.7 x ^ { 2 } + 0.3 xL(x)=0.7x2+0.3x

Evaluate ∫−22(x5+3x3−2x)dx\int _ { - 2 } ^ { 2 } \left( x ^ { 5 } + 3 x ^ { 3 } - 2 x \right) d x∫−22​(x5+3x3−2x)dx .

Evaluate ∫x2ex3dx\int x ^ { 2 } e ^ { x ^ { 3 } } d x∫x2ex3dx .

Determine the area of the region bounded by the line y = x and the curve y=x3y = x ^ { 3 }y=x3 .

Evaluate ∫232x−3x4dx\int _ { 2 } ^ { 3 } \frac { 2 x - 3 } { x ^ { 4 } } d x∫23​x42x−3​dx .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Determine the area of the region bounded by the curve $y = x ^ { 3 } - 3 x ^ { 2 } + x + 5$ and the line y = x + 5.

Evaluate $\int \frac { 2 x - 3 } { x ^ { 4 } } d x$ .

$\int ( x + 6 ) ^ { 10 } d x = \frac { ( x + 6 ) ^ { 11 } } { 11 } + C$

If $\int _ { a } ^ { b } f ( x ) d x = - 7$ and $\int _ { a } ^ { b } g ( x ) d x = 3$ , find $\int _ { b } ^ { a } [ f ( x ) - g ( x ) ] d x$ .

Solve the given initial value problem for y = f (x). $\frac { d y } { d x } = \frac { x - 5 } { x ^ { 2 } - 10 x + 8 }$ where y = 6 when x = 2

Specify the substitution you would choose to evaluate the integrals. $\int ( t + 4 ) e ^ { t ^ { 2 } + 8 t + 1 } d t$

A study indicates that x months from now the population of a certain city will be increasing at the rate of $( 4 + 3 x ) x ^ { - 1 / 2 }$ people per month. By how much will the population increase over the next 4 months?

$\int ( x - 6 ) ^ { 7 } d x = \frac { ( x - 6 ) ^ { 8 } } { 8 } + C$

Evaluate $\int \left( 5 x ^ { 2 } - e ^ { x } \right) d x$ .

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the curve $y = \sqrt { 16 - x ^ { 2 } }$ from x = -4 to x = 4.

Evaluate $\int _ { - 1 } ^ { 1 } x ^ { 15 } d x$ .

Find the average value of the function $f ( x ) = \frac { 2 x } { 2 x + 3 }$ over the interval [2, 6]. Round your answer to two decimal places.

Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. $L ( x ) = 0.7 x ^ { 2 } + 0.3 x$

Evaluate $\int _ { - 2 } ^ { 2 } \left( x ^ { 5 } + 3 x ^ { 3 } - 2 x \right) d x$ .

Evaluate $\int x ^ { 2 } e ^ { x ^ { 3 } } d x$ .

Determine the area of the region bounded by the line y = x and the curve $y = x ^ { 3 }$ .

Evaluate $\int _ { 2 } ^ { 3 } \frac { 2 x - 3 } { x ^ { 4 } } d x$ .