Question 1

Find the indefinite integral $\int v ^ { - 1 / 10 } d v$ and check your result by differentiation.

Accepted Answer

A)  $\frac { 9 v ^ { \frac { 9 } { 10 } } } { 10 }$ 
B)  $\frac { v ^ { \frac { 9 } { 10 } } } { 9 }$ 
C)  $\frac { 11 v ^ { \frac { 11 } { 10 } } } { 9 }$ 
D)  $\frac { 10 v ^ { \frac { 9 } { 10 } } } { 9 }$ 
E)  $\frac { v ^ { \frac { 11 } { 10 } } } { 11 }$ 
A)  $\frac { 9 v ^ { \frac { 9 } { 10 } } } { 10 }$ 
B)  $\frac { v ^ { \frac { 9 } { 10 } } } { 9 }$ 
C)  $\frac { 11 v ^ { \frac { 11 } { 10 } } } { 9 }$ 
D)  $\frac { 10 v ^ { \frac { 9 } { 10 } } } { 9 }$ 
E)  $\frac { v ^ { \frac { 11 } { 10 } } } { 11 }$

Question 2

Estimate the surface area of the oil spill shown in the figure using the Midpoint Rule.

Accepted Answer

A)  $\approx$ 481.6 $m i ^ { 2 }$ 
B)  $\approx$ 301.6 $m i ^ { 2 }$ 
C)  $\approx$ 311.6 $m i ^ { 2 }$ 
D)  $\approx$ 431.6 $m i ^ { 2 }$ 
E)  $\approx$ 381.6 $m i ^ { 2 }$ 
A)  $\approx$ 481.6 $m i ^ { 2 }$ 
B)  $\approx$ 301.6 $m i ^ { 2 }$ 
C)  $\approx$ 311.6 $m i ^ { 2 }$ 
D)  $\approx$ 431.6 $m i ^ { 2 }$ 
E)  $\approx$ 381.6 $m i ^ { 2 }$

Question 3

Evaluate the definite integral of the algebraic function. $\int _ { 2 } ^ { 4 } ( - 6 z + 4 ) d z$ Use a graphing utility to verify your results.

Accepted Answer

A) -12 
B) -56 
C) -36 
D) 8 
E) -28 
A) -12 
B) -56 
C) -36 
D) 8 
E) -28

Question 4

The graph of the derivative of a function is given below. Sketch the graphs of two functions that have the given derivative.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

Determine the graph whose area (the shaded region) is represented by the integral. $\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 4 x + 5 \right) - ( x + 1 ) d x$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 6

Evaluate the integral $\int \frac { 4 } { x ^ { 6 } } d x$ .

Accepted Answer

A)  $4 \ln \left( x ^ { 6 } \right) + C$ 
B)  $\frac { 4 } { 7 x ^ { 7 } } + C$ 
C)  $- \frac { 4 } { 5 } \ln \left( x ^ { 5 } \right) + C$ 
D)  $- \frac { 4 } { 7 x ^ { 7 } } + C$ 
E)  $- \frac { 4 } { 5 x ^ { 5 } } + C$ 
A)  $4 \ln \left( x ^ { 6 } \right) + C$ 
B)  $\frac { 4 } { 7 x ^ { 7 } } + C$ 
C)  $- \frac { 4 } { 5 } \ln \left( x ^ { 5 } \right) + C$ 
D)  $- \frac { 4 } { 7 x ^ { 7 } } + C$ 
E)  $- \frac { 4 } { 5 x ^ { 5 } } + C$

Question 7

Estimate the surface area of the golf green shown in the figure using the midpoint rule.

Accepted Answer

A) 966 
B) 161 
C) 1449 
D) 1550 
E) 234 
A) 966 
B) 161 
C) 1449 
D) 1550 
E) 234

Question 8

Use the Midpoint Rule with n = 4 to approximate the area of the following region. $f ( x ) = \frac { 1 } { x } , [ 1,5 ]$

Accepted Answer

A) 1.156 
B) 1.324 
C) 1.575 
D) 1.275 
E) 1.876 
A) 1.156 
B) 1.324 
C) 1.575 
D) 1.275 
E) 1.876

Question 9

Evaluate the definite integral of the algebraic function. $\int _ { 3 } ^ { 8 } \left( u ^ { \frac { 6 } { 5 } } - u ^ { \frac { 5 } { 6 } } \right) d u$ Use a graphing utility to verify your results.

Accepted Answer

A) 18.4007 
B) 29.6329 
C) 59.5941 
D) 38.9974 
E) 48.0336 
A) 18.4007 
B) 29.6329 
C) 59.5941 
D) 38.9974 
E) 48.0336

Question 10

Evaluate the integral $\int \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } \left( 40 x ^ { 4 } \right) d x$

Accepted Answer

A)  $\frac { 1 } { 6 } \left( 8 x ^ { 5 } + 9 \right) ^ { 6 } + C$ 
B)  $\frac { 1 } { 7 } \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } + C$ 
C)  $\frac { 1 } { 7 } \left( 8 x ^ { 5 } + 9 \right) ^ { 8 } + C$ 
D)  $\frac { 1 } { 8 } \left( 8 x ^ { 5 } + 9 \right) ^ { 8 } + C$ 
E)  $\frac { 1 } { 8 } \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } + C$ 
A)  $\frac { 1 } { 6 } \left( 8 x ^ { 5 } + 9 \right) ^ { 6 } + C$ 
B)  $\frac { 1 } { 7 } \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } + C$ 
C)  $\frac { 1 } { 7 } \left( 8 x ^ { 5 } + 9 \right) ^ { 8 } + C$ 
D)  $\frac { 1 } { 8 } \left( 8 x ^ { 5 } + 9 \right) ^ { 8 } + C$ 
E)  $\frac { 1 } { 8 } \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } + C$

Question 11

Use any basic integration formula or formulas to find the indefinite integral $\int e ^ { x } \sqrt { 2 - e ^ { x } } d x$ .

Accepted Answer

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

Question 12

Find the indefinite integral and check the result by differentiation. $\int \left( 16 x ^ { 3 } - 14 x + 5 \right) d x$

Accepted Answer

A)  $4 x ^ { 4 } - 7 x ^ { 2 } + 5 + C$ 
B)  $4 x ^ { 4 } - 7 x + 5 x$ 
C)  $12 x ^ { 3 } - 14 x ^ { 2 } + 5 x + C$ 
D)  $4 x ^ { 4 } - 7 x ^ { 2 } + 5 x + C$ 
E)  $4 x ^ { 4 } - 7 x ^ { 3 } + 5 x + C$ 
A)  $4 x ^ { 4 } - 7 x ^ { 2 } + 5 + C$ 
B)  $4 x ^ { 4 } - 7 x + 5 x$ 
C)  $12 x ^ { 3 } - 14 x ^ { 2 } + 5 x + C$ 
D)  $4 x ^ { 4 } - 7 x ^ { 2 } + 5 x + C$ 
E)  $4 x ^ { 4 } - 7 x ^ { 3 } + 5 x + C$

Question 13

Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of $f ( x ) = 3 x - x ^ { 4 }$ and the x-axis over the interval [0,1].

Accepted Answer

A) 1.7524 
B) 1.3126 
C) 1.5217 
D) 2.3103 
E) 1.3103 
A) 1.7524 
B) 1.3126 
C) 1.5217 
D) 2.3103 
E) 1.3103

Question 14

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 3 } ^ { 7 } 5 t ~d t$

Accepted Answer

A) -145 
B) 290 
C) 200 
D) 100 
E) 10 
A) -145 
B) 290 
C) 200 
D) 100 
E) 10

Question 15

An evergreen nursery sells a certain shrub after 4 years. The growth rate of the shrub is given by $d h / d t = 2.5 t + 6$ , where t is the time in years and h is the height in centimeters. The seedlings are 10 centimeters tall when planted (t = 0). How tall are the shrubs when they are sold?

Accepted Answer

A) 36 centimeters 
B) 40 centimeters 
C) 44 centimeters 
D) 54 centimeters 
E) 70 centimeters 
A) 36 centimeters 
B) 40 centimeters 
C) 44 centimeters 
D) 54 centimeters 
E) 70 centimeters

Question 16

Find the equation of the function f whose graph passes through the point $\left( 0 , \frac { 7 } { 3 } \right)$ and whose derivative is $f ^ { \prime } ( x ) = x \sqrt { 1 - x ^ { 2 } }$ .

Accepted Answer

A)  $f ( x ) = \frac { 1 } { 3 } \left[ 8 - \left( 1 - x ^ { 2 } \right) ^ { 1 / 2 } \right]$ 
B)  $f ( x ) = \frac { 1 } { 3 } \left[ 8 - \left( 1 - x ^ { 2 } \right) ^ { 3 / 2 } \right]$ 
C)  $f ( x ) = \frac { 1 } { 3 } \left[ 10 + \left( 1 - x ^ { 2 } \right) ^ { 2 / 3 } \right]$ 
D)  $f ( x ) = \frac { 1 } { 3 } \left[ 10 + \left( 1 - x ^ { 2 } \right) ^ { 1 / 2 } \right]$ 
E)  $f ( x ) = \frac { 1 } { 3 } \left[ 10 - \left( 1 - x ^ { 2 } \right) ^ { 2 / 3 } \right]$ 
A)  $f ( x ) = \frac { 1 } { 3 } \left[ 8 - \left( 1 - x ^ { 2 } \right) ^ { 1 / 2 } \right]$ 
B)  $f ( x ) = \frac { 1 } { 3 } \left[ 8 - \left( 1 - x ^ { 2 } \right) ^ { 3 / 2 } \right]$ 
C)  $f ( x ) = \frac { 1 } { 3 } \left[ 10 + \left( 1 - x ^ { 2 } \right) ^ { 2 / 3 } \right]$ 
D)  $f ( x ) = \frac { 1 } { 3 } \left[ 10 + \left( 1 - x ^ { 2 } \right) ^ { 1 / 2 } \right]$ 
E)  $f ( x ) = \frac { 1 } { 3 } \left[ 10 - \left( 1 - x ^ { 2 } \right) ^ { 2 / 3 } \right]$

Question 17

The demand function for a product is $p = 140 - 4 x$ , where p is the number of dollars and x is the number of units. If the equilibrium price is $\$ 40$ , what is the consumer's surplus?

Accepted Answer

A) $ $1175$ 
B) $ $1250$ 
C) $ $1375$ 
D) $ $1220$ 
E) $ $1340$ 
A) $ $1175$ 
B) $ $1250$ 
C) $ $1375$ 
D) $ $1220$ 
E) $ $1340$

Question 18

Find the area of the region bounded by the graphs of the algebraic functions. $\begin{array} { l } 
f ( x ) = x ^ { 2 } - 4 x \
g ( x ) = 0
\end{array}$

Accepted Answer

A)  $A = \frac { 17 } { 6 }$ 
B)  $A = \frac { 16 } { 3 }$ 
C)  $A = \frac { 11 } { 2 }$ 
D)  $A = \frac { 32 } { 3 }$ 
E)  $A = \frac { 3 } { 1 }$ 
A)  $A = \frac { 17 } { 6 }$ 
B)  $A = \frac { 16 } { 3 }$ 
C)  $A = \frac { 11 } { 2 }$ 
D)  $A = \frac { 32 } { 3 }$ 
E)  $A = \frac { 3 } { 1 }$

Question 19

Evaluate the integral $\int ( 7 x + 2 ) ^ { 1 / 5 } d x$

Accepted Answer

A)  $\frac { 5 } { 6 } ( 7 x + 2 ) ^ { 6 / 5 } + C$ 
B)  $\frac { 6 } { 5 } ( 7 x + 2 ) ^ { - 4 / 5 } + C$ 
C)  $\frac { 35 } { 6 } ( 7 x + 2 ) ^ { - 4 / 5 } + C$ 
D)  $\frac { 42 } { 5 } ( 7 x + 2 ) ^ { 6 / 5 } + C$ 
E)  $\frac { 5 } { 42 } ( 7 x + 2 ) ^ { 6 / 5 } + C$ 
A)  $\frac { 5 } { 6 } ( 7 x + 2 ) ^ { 6 / 5 } + C$ 
B)  $\frac { 6 } { 5 } ( 7 x + 2 ) ^ { - 4 / 5 } + C$ 
C)  $\frac { 35 } { 6 } ( 7 x + 2 ) ^ { - 4 / 5 } + C$ 
D)  $\frac { 42 } { 5 } ( 7 x + 2 ) ^ { 6 / 5 } + C$ 
E)  $\frac { 5 } { 42 } ( 7 x + 2 ) ^ { 6 / 5 } + C$

Question 20

Find the supply function $x = f ( p )$ that satisfies $\frac { d x } { d p } = p \sqrt { p ^ { 2 } - 25 }$ and the initial condition x = 700 when $p = \$ 13$ .

Accepted Answer

A)  $x = \frac { 1 } { 3 } \left( p ^ { 2 } - 25 \right) ^ { 3 / 2 } + 124$ 
B)  $x = \frac { 1 } { 3 } \left( p ^ { 2 } - 25 \right) ^ { 1 / 2 } + 696$ 
C)  $x = \frac { 1 } { 3 } ( p - 5 ) + 124$ 
D)  $x = \frac { 1 } { 5 } \left( p ^ { 2 } - 25 \right) ^ { 3 / 2 } + 127$ 
E)  $x = \frac { 1 } { p - 1 } \left( p ^ { 2 } - 25 \right) ^ { 1 / 2 } + 699$ 
A)  $x = \frac { 1 } { 3 } \left( p ^ { 2 } - 25 \right) ^ { 3 / 2 } + 124$ 
B)  $x = \frac { 1 } { 3 } \left( p ^ { 2 } - 25 \right) ^ { 1 / 2 } + 696$ 
C)  $x = \frac { 1 } { 3 } ( p - 5 ) + 124$ 
D)  $x = \frac { 1 } { 5 } \left( p ^ { 2 } - 25 \right) ^ { 3 / 2 } + 127$ 
E)  $x = \frac { 1 } { p - 1 } \left( p ^ { 2 } - 25 \right) ^ { 1 / 2 } + 699$

Find the indefinite integral $\int v ^ { - 1 / 10 } d v$ and check your result by differentiation.

Estimate the surface area of the oil spill shown in the figure using the Midpoint Rule.

Evaluate the definite integral of the algebraic function. $\int _ { 2 } ^ { 4 } ( - 6 z + 4 ) d z$ Use a graphing utility to verify your results.

The graph of the derivative of a function is given below. Sketch the graphs of two functions that have the given derivative.

Determine the graph whose area (the shaded region) is represented by the integral. $\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 4 x + 5 \right) - ( x + 1 ) d x$

Evaluate the integral $\int \frac { 4 } { x ^ { 6 } } d x$ .

Estimate the surface area of the golf green shown in the figure using the midpoint rule.

Use the Midpoint Rule with n = 4 to approximate the area of the following region. $f ( x ) = \frac { 1 } { x } , [ 1,5 ]$ $Use the Midpoint Rule with n = 4 to approximate the area of the following region. f ( x ) = \frac { 1 } { x } , [ 1,5 ]$

Evaluate the definite integral of the algebraic function. $\int _ { 3 } ^ { 8 } \left( u ^ { \frac { 6 } { 5 } } - u ^ { \frac { 5 } { 6 } } \right) d u$ Use a graphing utility to verify your results.

Evaluate the integral $\int \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } \left( 40 x ^ { 4 } \right) d x$

Use any basic integration formula or formulas to find the indefinite integral $\int e ^ { x } \sqrt { 2 - e ^ { x } } d x$ .

Find the indefinite integral and check the result by differentiation. $\int \left( 16 x ^ { 3 } - 14 x + 5 \right) d x$

Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of $f ( x ) = 3 x - x ^ { 4 }$ and the x-axis over the interval [0,1].

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 3 } ^ { 7 } 5 t ~d t$

An evergreen nursery sells a certain shrub after 4 years. The growth rate of the shrub is given by $d h / d t = 2.5 t + 6$ , where t is the time in years and h is the height in centimeters. The seedlings are 10 centimeters tall when planted (t = 0). How tall are the shrubs when they are sold?

Find the equation of the function f whose graph passes through the point $\left( 0 , \frac { 7 } { 3 } \right)$ and whose derivative is $f ^ { \prime } ( x ) = x \sqrt { 1 - x ^ { 2 } }$ .

The demand function for a product is $p = 140 - 4 x$ , where p is the number of dollars and x is the number of units. If the equilibrium price is $\$ 40$ , what is the consumer's surplus?

Find the area of the region bounded by the graphs of the algebraic functions. f(x)=-4x g(x)=0

Evaluate the integral $\int ( 7 x + 2 ) ^ { 1 / 5 } d x$

Find the supply function $x = f ( p )$ that satisfies $\frac { d x } { d p } = p \sqrt { p ^ { 2 } - 25 }$ and the initial condition x = 700 when $p = \$ 13$ .

Fundamental Concepts of Algebra

Equations and Inequalities

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Exam 12: Integration and Its Applications

Find the indefinite integral ∫v−1/10dv\int v ^ { - 1 / 10 } d v∫v−1/10dv and check your result by differentiation.

Estimate the surface area of the oil spill shown in the figure using the Midpoint Rule.

Evaluate the definite integral of the algebraic function. ∫24(−6z+4)dz\int _ { 2 } ^ { 4 } ( - 6 z + 4 ) d z∫24​(−6z+4)dz Use a graphing utility to verify your results.

The graph of the derivative of a function is given below. Sketch the graphs of two functions that have the given derivative.

Determine the graph whose area (the shaded region) is represented by the integral. ∫14(x2−4x+5)−(x+1)dx\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 4 x + 5 \right) - ( x + 1 ) d x∫14​(x2−4x+5)−(x+1)dx

Evaluate the integral ∫4x6dx\int \frac { 4 } { x ^ { 6 } } d x∫x64​dx .

Estimate the surface area of the golf green shown in the figure using the midpoint rule.

Use the Midpoint Rule with n = 4 to approximate the area of the following region. f(x)=1x,[1,5]f ( x ) = \frac { 1 } { x } , [ 1,5 ]f(x)=x1​,[1,5]

Evaluate the definite integral of the algebraic function. ∫38(u65−u56)du\int _ { 3 } ^ { 8 } \left( u ^ { \frac { 6 } { 5 } } - u ^ { \frac { 5 } { 6 } } \right) d u∫38​(u56​−u65​)du Use a graphing utility to verify your results.

Evaluate the integral ∫(8x5+9)7(40x4)dx\int \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } \left( 40 x ^ { 4 } \right) d x∫(8x5+9)7(40x4)dx

Use any basic integration formula or formulas to find the indefinite integral ∫ex2−exdx\int e ^ { x } \sqrt { 2 - e ^ { x } } d x∫ex2−ex​dx .

Find the indefinite integral and check the result by differentiation. ∫(16x3−14x+5)dx\int \left( 16 x ^ { 3 } - 14 x + 5 \right) d x∫(16x3−14x+5)dx

Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f(x)=3x−x4f ( x ) = 3 x - x ^ { 4 }f(x)=3x−x4 and the x-axis over the interval [0,1].

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. ∫375t dt\int _ { 3 } ^ { 7 } 5 t ~d t∫37​5t dt

Find the equation of the function f whose graph passes through the point (0,73)\left( 0 , \frac { 7 } { 3 } \right)(0,37​) and whose derivative is f′(x)=x1−x2f ^ { \prime } ( x ) = x \sqrt { 1 - x ^ { 2 } }f′(x)=x1−x2​ .

The demand function for a product is p=140−4xp = 140 - 4 xp=140−4x , where p is the number of dollars and x is the number of units. If the equilibrium price is $40\$ 40$40 , what is the consumer's surplus?

Find the area of the region bounded by the graphs of the algebraic functions. f(x)=-4x g(x)=0

Evaluate the integral ∫(7x+2)1/5dx\int ( 7 x + 2 ) ^ { 1 / 5 } d x∫(7x+2)1/5dx

Find the supply function x=f(p)x = f ( p )x=f(p) that satisfies dxdp=pp2−25\frac { d x } { d p } = p \sqrt { p ^ { 2 } - 25 }dpdx​=pp2−25​ and the initial condition x = 700 when p=$13p = \$ 13p=$13 .

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

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Filters

Find the indefinite integral $\int v ^ { - 1 / 10 } d v$ and check your result by differentiation.

Evaluate the definite integral of the algebraic function. $\int _ { 2 } ^ { 4 } ( - 6 z + 4 ) d z$ Use a graphing utility to verify your results.

Determine the graph whose area (the shaded region) is represented by the integral. $\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 4 x + 5 \right) - ( x + 1 ) d x$

Evaluate the integral $\int \frac { 4 } { x ^ { 6 } } d x$ .

Use the Midpoint Rule with n = 4 to approximate the area of the following region. $f ( x ) = \frac { 1 } { x } , [ 1,5 ]$ $Use the Midpoint Rule with n = 4 to approximate the area of the following region. f ( x ) = \frac { 1 } { x } , [ 1,5 ]$

Evaluate the definite integral of the algebraic function. $\int _ { 3 } ^ { 8 } \left( u ^ { \frac { 6 } { 5 } } - u ^ { \frac { 5 } { 6 } } \right) d u$ Use a graphing utility to verify your results.

Evaluate the integral $\int \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } \left( 40 x ^ { 4 } \right) d x$

Use any basic integration formula or formulas to find the indefinite integral $\int e ^ { x } \sqrt { 2 - e ^ { x } } d x$ .

Find the indefinite integral and check the result by differentiation. $\int \left( 16 x ^ { 3 } - 14 x + 5 \right) d x$

Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of $f ( x ) = 3 x - x ^ { 4 }$ and the x-axis over the interval [0,1].

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 3 } ^ { 7 } 5 t ~d t$

Find the equation of the function f whose graph passes through the point $\left( 0 , \frac { 7 } { 3 } \right)$ and whose derivative is $f ^ { \prime } ( x ) = x \sqrt { 1 - x ^ { 2 } }$ .

The demand function for a product is $p = 140 - 4 x$ , where p is the number of dollars and x is the number of units. If the equilibrium price is $\$ 40$ , what is the consumer's surplus?

Evaluate the integral $\int ( 7 x + 2 ) ^ { 1 / 5 } d x$

Find the supply function $x = f ( p )$ that satisfies $\frac { d x } { d p } = p \sqrt { p ^ { 2 } - 25 }$ and the initial condition x = 700 when $p = \$ 13$ .