Question 1

If $\int _ { - 4 } ^ { 10 } f ( x ) d x = - 10$ and $\int _ { 8 } ^ { 10 } f ( x ) d x = 5$ , find $\int _ { - 4 } ^ { 8 } f ( x ) d x$ .

Accepted Answer

The answer of If \(\int _ { - 4 }...

Question 2

It is estimated that t days from now a farmer's crop will be increasing at the rate of $0.3 t ^ { 2 } + 0.6 t + 1$ bushels per day. By how much will the value of the crop increase during the next 8 days if the market price remains fixed at $6 per bushel?

Accepted Answer

A)  $422.40 
B)  $470.40 
C)  $500.40 
D)  $100.80 
A)  $422.40 
B)  $470.40 
C)  $500.40 
D)  $100.80

Question 3

Evaluate $\int _ { 0 } ^ { 1 } \left( 9 x ^ { 3 } - 6 x ^ { 2 } + 7 \right) d x$ .

Accepted Answer

The answer of Evaluate \(\int _ { 0 } ^...

Question 4

For the demand function $D ( q ) = 3 \left( 80 - q ^ { 2 } \right)$ dollars per unit, find the total amount of money consumer's are willing to spend when q = 5 units.

Accepted Answer

A)  $1,002 
B)  $1,053 
C)  $1,061 
D)  $1,075 
A)  $1,002 
B)  $1,053 
C)  $1,061 
D)  $1,075

Question 5

Water flows into a tank at the rate of $\sqrt { 4 t + 5 }$ ft3/min. If the tank is empty when t = 0, how much water does it contain 5 minutes later? Express the answer to two decimal places.

Accepted Answer

Question 6

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

Accepted Answer

The answer of Evaluate \(\int \frac { 3 } {...

Question 7

A certain nuclear power plant produces nuclear waste in the form of strontium-90 at the constant rate of 600 pounds per year. The waste decays exponentially with a half-life of 28 years. How much of the radioactive waste from the nuclear plant will be present after 110 years? [Hint: Think of this as a survival and renewal problem.] Round to two decimal places.

Accepted Answer

A)  25,828.99 pounds 
B)  11,322.78 pounds 
C)  22,645.56 pounds 
D)  12,066.37 pounds 
A)  25,828.99 pounds 
B)  11,322.78 pounds 
C)  22,645.56 pounds 
D)  12,066.37 pounds

Question 8

The area of the region bounded by the curve $y = \frac { x ^ { 3 } } { 16 }$ and the line y = x is 9.

Accepted Answer

A) True 
 B)False

Question 9

Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. $L ( x ) = 0.7 x ^ { 2 } + 0.1 x$

Accepted Answer

The answer of Find the Gini index for the given...

Question 10

Given a consumer's demand function, $D ( q ) = \frac { 400 } { 0.7 q + 8 }$ dollars per unit, find the total amount of money consumers are willing to spend to get 10 units of the commodity.

Accepted Answer

A)  $0.63 
B)  $251.44 
C)  $179.60 
D)  $359.20 
A)  $0.63 
B)  $251.44 
C)  $179.60 
D)  $359.20

Question 11

$\int \left( \frac { 9 } { x ^ { 2 } } - \frac { 2 } { x ^ { 3 } } + \frac { 8 } { \sqrt { x } } \right) d x = - \frac { 9 } { x } + \frac { 1 } { x ^ { 2 } } + 8 \ln | \sqrt { x } | + C$

Accepted Answer

A) True 
 B)False

Question 12

The average value of $f ( x ) = x e ^ { - x }$ over the interval 0 $\le$  x $\le$  3 is $\frac { 4 e ^ { - 3 } } { 3 }$ .

Accepted Answer

A) True 
 B)False

Question 13

Determine the area of the region bounded by the curves $y = e ^ { x }$ and $y = e ^ { - x }$ between x = 0 and x = ln 2.

Accepted Answer

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

Question 14

Compute the area of the finite region bounded by the graphs of the functions $y = 2 x ^ { 2 }$ , and $y = - 2 x ^ { 2 } - 8 x + 32$ .

Accepted Answer

The answer of Compute the area of the finite region...

Question 15

Evaluate $\int x \sqrt { x ^ { 2 } + 9 } d x$ .

Accepted Answer

A)  $\frac { \left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } } { 3 } + C$ 
B)  $\left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } + C$ 
C)  $\frac { x ^ { 3 } } { 3 } + 9 x + C$ 
D)  $\frac { 3 \left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } } { 4 } + C$ 
A)  $\frac { \left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } } { 3 } + C$ 
B)  $\left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } + C$ 
C)  $\frac { x ^ { 3 } } { 3 } + 9 x + C$ 
D)  $\frac { 3 \left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } } { 4 } + C$

Question 16

$\int ( x + 3 ) ^ { - 5 } d x = - \frac { ( x + 3 ) ^ { - 6 } } { 6 } + C$

Accepted Answer

A) True 
 B)False

Question 17

Evaluate $\int \left( \frac { 1 } { 3 } x - 9 \right) ^ { 2 / 3 } d x$ .

Accepted Answer

A)  $\frac { x ^ { 2 } } { 6 } - 9 x + C$ 
B)  $\frac { 3 } { 5 } \left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$ 
C)  $\frac { 9 } { 5 } \left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$ 
D)  $\left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$ 
A)  $\frac { x ^ { 2 } } { 6 } - 9 x + C$ 
B)  $\frac { 3 } { 5 } \left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$ 
C)  $\frac { 9 } { 5 } \left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$ 
D)  $\left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$

Question 18

Evaluate $\int x ^ { 6 } \sqrt { x ^ { 7 } + 8 } d x$ .

Accepted Answer

A)  $\frac { 2 \left( x ^ { 7 } + 8 \right) ^ { 3 / 2 } } { 21 } + C$ 
B)  $\left( x ^ { 7 } + 8 \right) ^ { 3 / 2 } + C$ 
C)  $C \left( x ^ { 7 } + 8 \right) ^ { 3 / 2 }$ 
D)  $x ^ { 7 } + 8 + C$ 
A)  $\frac { 2 \left( x ^ { 7 } + 8 \right) ^ { 3 / 2 } } { 21 } + C$ 
B)  $\left( x ^ { 7 } + 8 \right) ^ { 3 / 2 } + C$ 
C)  $C \left( x ^ { 7 } + 8 \right) ^ { 3 / 2 }$ 
D)  $x ^ { 7 } + 8 + C$

Question 19

Evaluate: $\int x ^ { 3 } ( 2 x + 5 - 3 \sqrt { x } ) d x$

Accepted Answer

A)  $\frac { x ^ { 4 } } { 4 } \left( x ^ { 2 } + 5 x - 2 | x | \sqrt { x } \right) + C$ 
B)  $\frac { 2 x ^ { 5 } } { 5 } + 5 x ^ { 3 } - \frac { 6 } { 5 } x ^ { 2 } \sqrt { x } + C$ 
C)  $8 x ^ { 3 } + 15 x ^ { 2 } - \frac { 21 } { 2 } x ^ { 2 } \sqrt { x } + C$ 
D)  $\frac { 2 x ^ { 5 } } { 5 } + \frac { 5 x ^ { 4 } } { 4 } - \frac { 2 } { 3 } x ^ { 4 } \sqrt { x } + C$ 
A)  $\frac { x ^ { 4 } } { 4 } \left( x ^ { 2 } + 5 x - 2 | x | \sqrt { x } \right) + C$ 
B)  $\frac { 2 x ^ { 5 } } { 5 } + 5 x ^ { 3 } - \frac { 6 } { 5 } x ^ { 2 } \sqrt { x } + C$ 
C)  $8 x ^ { 3 } + 15 x ^ { 2 } - \frac { 21 } { 2 } x ^ { 2 } \sqrt { x } + C$ 
D)  $\frac { 2 x ^ { 5 } } { 5 } + \frac { 5 x ^ { 4 } } { 4 } - \frac { 2 } { 3 } x ^ { 4 } \sqrt { x } + C$

Question 20

Find the average value of the function $f ( x ) = \frac { 2 x } { 2 x + 1 }$ over the interval [2, 6]. Round your answer to two decimal places.

Accepted Answer

The answer of Find the average value of the function...

If $\int _ { - 4 } ^ { 10 } f ( x ) d x = - 10$ and $\int _ { 8 } ^ { 10 } f ( x ) d x = 5$ , find $\int _ { - 4 } ^ { 8 } f ( x ) d x$ .

It is estimated that t days from now a farmer's crop will be increasing at the rate of $0.3 t ^ { 2 } + 0.6 t + 1$ bushels per day. By how much will the value of the crop increase during the next 8 days if the market price remains fixed at $6 per bushel?

Evaluate $\int _ { 0 } ^ { 1 } \left( 9 x ^ { 3 } - 6 x ^ { 2 } + 7 \right) d x$ .

For the demand function $D ( q ) = 3 \left( 80 - q ^ { 2 } \right)$ dollars per unit, find the total amount of money consumer's are willing to spend when q = 5 units.

Water flows into a tank at the rate of $\sqrt { 4 t + 5 }$ ft³/min. If the tank is empty when t = 0, how much water does it contain 5 minutes later? Express the answer to two decimal places.

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

The area of the region bounded by the curve $y = \frac { x ^ { 3 } } { 16 }$ and the line y = x is 9.

Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. $L ( x ) = 0.7 x ^ { 2 } + 0.1 x$

Given a consumer's demand function, $D ( q ) = \frac { 400 } { 0.7 q + 8 }$ dollars per unit, find the total amount of money consumers are willing to spend to get 10 units of the commodity.

$\int \left( \frac { 9 } { x ^ { 2 } } - \frac { 2 } { x ^ { 3 } } + \frac { 8 } { \sqrt { x } } \right) d x = - \frac { 9 } { x } + \frac { 1 } { x ^ { 2 } } + 8 \ln | \sqrt { x } | + C$

The average value of $f ( x ) = x e ^ { - x }$ over the interval 0 $\le$ x $\le$ 3 is $\frac { 4 e ^ { - 3 } } { 3 }$ .

Determine the area of the region bounded by the curves $y = e ^ { x }$ and $y = e ^ { - x }$ between x = 0 and x = ln 2.

Compute the area of the finite region bounded by the graphs of the functions $y = 2 x ^ { 2 }$ , and $y = - 2 x ^ { 2 } - 8 x + 32$ .

Evaluate $\int x \sqrt { x ^ { 2 } + 9 } d x$ .

$\int ( x + 3 ) ^ { - 5 } d x = - \frac { ( x + 3 ) ^ { - 6 } } { 6 } + C$

Evaluate $\int \left( \frac { 1 } { 3 } x - 9 \right) ^ { 2 / 3 } d x$ .

Evaluate $\int x ^ { 6 } \sqrt { x ^ { 7 } + 8 } d x$ .

Evaluate: $\int x ^ { 3 } ( 2 x + 5 - 3 \sqrt { x } ) d x$

Find the average value of the function $f ( x ) = \frac { 2 x } { 2 x + 1 }$ over the interval [2, 6]. Round your answer to two decimal places.

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

If ∫−410f(x)dx=−10\int _ { - 4 } ^ { 10 } f ( x ) d x = - 10∫−410​f(x)dx=−10 and ∫810f(x)dx=5\int _ { 8 } ^ { 10 } f ( x ) d x = 5∫810​f(x)dx=5 , find ∫−48f(x)dx\int _ { - 4 } ^ { 8 } f ( x ) d x∫−48​f(x)dx .

It is estimated that t days from now a farmer's crop will be increasing at the rate of 0.3t2+0.6t+10.3 t ^ { 2 } + 0.6 t + 10.3t2+0.6t+1 bushels per day. By how much will the value of the crop increase during the next 8 days if the market price remains fixed at $6 per bushel?

Evaluate ∫01(9x3−6x2+7)dx\int _ { 0 } ^ { 1 } \left( 9 x ^ { 3 } - 6 x ^ { 2 } + 7 \right) d x∫01​(9x3−6x2+7)dx .

For the demand function D(q)=3(80−q2)D ( q ) = 3 \left( 80 - q ^ { 2 } \right)D(q)=3(80−q2) dollars per unit, find the total amount of money consumer's are willing to spend when q = 5 units.

Water flows into a tank at the rate of 4t+5\sqrt { 4 t + 5 }4t+5​ ft3/min. If the tank is empty when t = 0, how much water does it contain 5 minutes later? Express the answer to two decimal places.

Evaluate ∫3x−4dx\int \frac { 3 } { x - 4 } d x∫x−43​dx .

The area of the region bounded by the curve y=x316y = \frac { x ^ { 3 } } { 16 }y=16x3​ and the line y = x is 9.

Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. L(x)=0.7x2+0.1xL ( x ) = 0.7 x ^ { 2 } + 0.1 xL(x)=0.7x2+0.1x

Given a consumer's demand function, D(q)=4000.7q+8D ( q ) = \frac { 400 } { 0.7 q + 8 }D(q)=0.7q+8400​ dollars per unit, find the total amount of money consumers are willing to spend to get 10 units of the commodity.

∫(9x2−2x3+8x)dx=−9x+1x2+8ln⁡∣x∣+C\int \left( \frac { 9 } { x ^ { 2 } } - \frac { 2 } { x ^ { 3 } } + \frac { 8 } { \sqrt { x } } \right) d x = - \frac { 9 } { x } + \frac { 1 } { x ^ { 2 } } + 8 \ln | \sqrt { x } | + C∫(x29​−x32​+x​8​)dx=−x9​+x21​+8ln∣x​∣+C

The average value of f(x)=xe−xf ( x ) = x e ^ { - x }f(x)=xe−x over the interval 0 ≤\le≤ x ≤\le≤ 3 is 4e−33\frac { 4 e ^ { - 3 } } { 3 }34e−3​ .

Determine the area of the region bounded by the curves y=exy = e ^ { x }y=ex and y=e−xy = e ^ { - x }y=e−x between x = 0 and x = ln 2.

Compute the area of the finite region bounded by the graphs of the functions y=2x2y = 2 x ^ { 2 }y=2x2 , and y=−2x2−8x+32y = - 2 x ^ { 2 } - 8 x + 32y=−2x2−8x+32 .

Evaluate ∫xx2+9dx\int x \sqrt { x ^ { 2 } + 9 } d x∫xx2+9​dx .

∫(x+3)−5dx=−(x+3)−66+C\int ( x + 3 ) ^ { - 5 } d x = - \frac { ( x + 3 ) ^ { - 6 } } { 6 } + C∫(x+3)−5dx=−6(x+3)−6​+C

Evaluate ∫(13x−9)2/3dx\int \left( \frac { 1 } { 3 } x - 9 \right) ^ { 2 / 3 } d x∫(31​x−9)2/3dx .

Evaluate ∫x6x7+8dx\int x ^ { 6 } \sqrt { x ^ { 7 } + 8 } d x∫x6x7+8​dx .

Evaluate: ∫x3(2x+5−3x)dx\int x ^ { 3 } ( 2 x + 5 - 3 \sqrt { x } ) d x∫x3(2x+5−3x​)dx

Find the average value of the function f(x)=2x2x+1f ( x ) = \frac { 2 x } { 2 x + 1 }f(x)=2x+12x​ over the interval [2, 6]. Round your answer to two decimal places.

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

If $\int _ { - 4 } ^ { 10 } f ( x ) d x = - 10$ and $\int _ { 8 } ^ { 10 } f ( x ) d x = 5$ , find $\int _ { - 4 } ^ { 8 } f ( x ) d x$ .

It is estimated that t days from now a farmer's crop will be increasing at the rate of $0.3 t ^ { 2 } + 0.6 t + 1$ bushels per day. By how much will the value of the crop increase during the next 8 days if the market price remains fixed at $6 per bushel?

Evaluate $\int _ { 0 } ^ { 1 } \left( 9 x ^ { 3 } - 6 x ^ { 2 } + 7 \right) d x$ .

For the demand function $D ( q ) = 3 \left( 80 - q ^ { 2 } \right)$ dollars per unit, find the total amount of money consumer's are willing to spend when q = 5 units.

Water flows into a tank at the rate of $\sqrt { 4 t + 5 }$ ft³/min. If the tank is empty when t = 0, how much water does it contain 5 minutes later? Express the answer to two decimal places.

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

The area of the region bounded by the curve $y = \frac { x ^ { 3 } } { 16 }$ and the line y = x is 9.

Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. $L ( x ) = 0.7 x ^ { 2 } + 0.1 x$

Given a consumer's demand function, $D ( q ) = \frac { 400 } { 0.7 q + 8 }$ dollars per unit, find the total amount of money consumers are willing to spend to get 10 units of the commodity.

$\int \left( \frac { 9 } { x ^ { 2 } } - \frac { 2 } { x ^ { 3 } } + \frac { 8 } { \sqrt { x } } \right) d x = - \frac { 9 } { x } + \frac { 1 } { x ^ { 2 } } + 8 \ln | \sqrt { x } | + C$

The average value of $f ( x ) = x e ^ { - x }$ over the interval 0 $\le$ x $\le$ 3 is $\frac { 4 e ^ { - 3 } } { 3 }$ .

Determine the area of the region bounded by the curves $y = e ^ { x }$ and $y = e ^ { - x }$ between x = 0 and x = ln 2.

Compute the area of the finite region bounded by the graphs of the functions $y = 2 x ^ { 2 }$ , and $y = - 2 x ^ { 2 } - 8 x + 32$ .

Evaluate $\int x \sqrt { x ^ { 2 } + 9 } d x$ .

$\int ( x + 3 ) ^ { - 5 } d x = - \frac { ( x + 3 ) ^ { - 6 } } { 6 } + C$

Evaluate $\int \left( \frac { 1 } { 3 } x - 9 \right) ^ { 2 / 3 } d x$ .

Evaluate $\int x ^ { 6 } \sqrt { x ^ { 7 } + 8 } d x$ .

Evaluate: $\int x ^ { 3 } ( 2 x + 5 - 3 \sqrt { x } ) d x$

Find the average value of the function $f ( x ) = \frac { 2 x } { 2 x + 1 }$ over the interval [2, 6]. Round your answer to two decimal places.