Question 1

Express $\int _ { 4 } ^ { 64 } [ \sin ( \pi x ) ] d x$ as the limit of a sum.

Accepted Answer

The answer of Express \(\int _ { 4 } ^...

Question 2

Let f (x) = $2 ^ { 6 x }$ on the interval [0,1]. Let the interval be divided into three equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 3 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i }$ if each $x _ { i } ^ { * }$ is the midpoint of its subinterval.

Accepted Answer

A)  $\frac { 13 } { 3 }$ 
B) 21 
C) 7 
D) 28 
E) 8
F)42
G)14
H)84 
A)  $\frac { 13 } { 3 }$ 
B) 21 
C) 7 
D) 28 
E) 8
F)42
G)14
H)84

Question 3

Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } t ^ { 2 } d t \text {. Find the value of } f ^ { \prime } ( 1 ) \text {. }$

Accepted Answer

A) 4 
B) 6 
C) 8 
D) 0 
E) 2
F)1
G)3
H)5 
A) 4 
B) 6 
C) 8 
D) 0 
E) 2
F)1
G)3
H)5

Question 4

Let $f ( x ) = \int _ { 1 } ^ { x ^ { 2 } } \frac { 4 - u } { \sin u + 2 } d u$ Find the interval(s) on which $f ( x )$ is increasing.

Accepted Answer

The answer of Let \(f ( x ) = \int...

Question 5

Let f be a twice differentiable function such that f(0) = 5, f(3) = 1, and $f ^ { \prime }$ (3) = $- 2$ . Determine the value of $\int _ { 0 } ^ { 3 } x \cdot f ^ { \prime \prime } ( x ) d x$ .

Accepted Answer

The answer of Let f be a twice differentiable function...

Question 6

Find the value of the integral $\int _ { 0 } ^ { x / 3 } \sec x \tan x ( 1 + \sec x ) d x$

Accepted Answer

A) 4 
B)  $\frac { 5 } { 2 }$ 
C) 3 
D)  $\frac { 11 } { 2 }$ 
E)  $\frac { 9 } { 2 }$ 
F)2
G) $\frac { 7 } { 2 }$  
H)5 
A) 4 
B)  $\frac { 5 } { 2 }$ 
C) 3 
D)  $\frac { 11 } { 2 }$ 
E)  $\frac { 9 } { 2 }$ 
F)2
G) $\frac { 7 } { 2 }$  
H)5

Question 7

Below is a table of values for a continuous function f. $\begin{array} { | c | c | c | c | c | c | } 
\hline x & - 1 & - 0.5 & 0 & 0.5 & 1 \
\hline f ( x ) &2.0 &1.5 &-1.0 &0.5 &-3.0 \
\hline
\end{array}$
(a) Use the Trapezoidal Rule with n = 4 to approximate $\int _ { - 1 } ^ { 1 } f ( x ) d x$ 
(b) Use Simpson's Rule with n = 4 to approximate $\int _ { - 1 } ^ { 1 } f ( x ) d x$

Accepted Answer

The answer of Below is a table of values for...

Question 8

If $\int _ { 0 } ^ { 4 } f ( x ) d x = 5 \text { and } \int _ { 0 } ^ { 4 } g ( x ) d x = 2 \text {, find the value of } \int _ { 0 } ^ { 4 } f ( x ) g ( x ) d x \text {. }$

Accepted Answer

A) 10 
B) 6 
C) 1 
D) 8 
E) 4
F)3
G)2
H)Cannot be determined 
A) 10 
B) 6 
C) 1 
D) 8 
E) 4
F)3
G)2
H)Cannot be determined

Question 9

Evaluate the following integrals:
(a) $\int x ^ { 3 } e ^ { \left( x ^ { 4 } + 1 \right) } d x$ (b) $\int \frac { e ^ { ( \sqrt { x } ) } } { \sqrt { x } } d x$ (c) $\int \frac { e ^ { ( 1 / x ) } } { x ^ { 2 } } d x$ (d) $\int \frac { e ^ { ( \arctan x ) } } { 1 + x ^ { 2 } } d x$

Accepted Answer

The answer of Evaluate the following integrals:
(a) \(\int x ^...

Question 10

If $\int _ { 3 } ^ { b } 3 x ^ { 2 } d x = 37$ find the value of b.

Accepted Answer

A) 4 
B) 5 
C) 6 
D) 7 
E) 8
F)9
G)10
H)11 
A) 4 
B) 5 
C) 6 
D) 7 
E) 8
F)9
G)10
H)11

Question 11

Use the Comparison Theorem to determine whether $\int _ { 0 } ^ { \infty } \frac { 1 } { \sqrt { x } + 1 } d x$ is convergent or divergent. Justify your answer.

Accepted Answer

Divergent.

Question 12

Estimate $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } } d x$ using the Midpoint Rule with n = 4. Then use the error bound $\left| E _ { M } \right| \leq \frac { k ( b - a ) ^ { 3 } } { 24 n ^ { 2 } }$ to estimate the accuracy.

Accepted Answer

The answer of Estimate \(\int _ { 1 } ^...

Question 13

Let $f ( x ) = \int _ { 0 } ^ { 5 } t ^ { 4 } d t . \text { Find } f ^ { \prime } ( x )$

Accepted Answer

A) 0 
B)  $2 x ^ { 6 }$ 
C)  $18 x ^ { 8 }$ 
D)  $x ^ { 6 }$ 
E)  $2 x ^ { 2 }$ 
F) $x ^ { 4 }$  
G) $12 x ^ { 5 }$  
H) $2 x ^ { 9 }$ 
A) 0 
B)  $2 x ^ { 6 }$ 
C)  $18 x ^ { 8 }$ 
D)  $x ^ { 6 }$ 
E)  $2 x ^ { 2 }$ 
F) $x ^ { 4 }$  
G) $12 x ^ { 5 }$  
H) $2 x ^ { 9 }$

Question 14

Suppose f (x) is a increasing function on the interval $1 \leq x \leq 4$ . If you estimate $\int _ { 1 } ^ { 4 } f ( x ) d x$ using right endpoints, explain why the estimate is an overestimate or an underestimate.

Accepted Answer

The estimate using right endpo

Question 15

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.(a) $\int _ { 0 } ^ { 2 } \frac { 1 } { ( x - 1 ) ^ { 2 } } d x$ (b) $\int _ { 0 } ^ { 3 } \frac { d x } { ( x - 1 ) ^ { 2 / 3 } } d x$ (c) $\int _ { 0 } ^ { x } \sec ^ { 2 } t d t$

Accepted Answer

(a) Diverg

Question 16

Find the value of the integral $\int _ { 0 } ^ { x / 2 } \cos ^ { 3 } x d x$

Accepted Answer

A) 1 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D) 2 
E) 3
F) $\frac { \pi } { 2 }$ 
G) $\frac { 1 } { 6 }$ 
H) $\frac { \pi } { 2 } - \frac { \pi ^ { 3 } } { 24 }$ 
A) 1 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D) 2 
E) 3
F) $\frac { \pi } { 2 }$ 
G) $\frac { 1 } { 6 }$ 
H) $\frac { \pi } { 2 } - \frac { \pi ^ { 3 } } { 24 }$

Question 17

Find the value of the integral $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x ^ { 3 } } d x$

Accepted Answer

A)  $\frac { - e } { 2 }$ 
B)  $\frac { e } { 2 }$ 
C) $\frac { 1 - e ^ { - 1 } } { 2 }$ 
D)  $e$ 
E) $- e$ 
F) $\frac { 1 - e ^ { - 1 } } { 3 }$

G) $\frac { e ^ { - 1 } } { 2 }$

H) $e ^ { - 1 }$ 
A)  $\frac { - e } { 2 }$ 
B)  $\frac { e } { 2 }$ 
C) $\frac { 1 - e ^ { - 1 } } { 2 }$ 
D)  $e$ 
E) $- e$ 
F) $\frac { 1 - e ^ { - 1 } } { 3 }$

G) $\frac { e ^ { - 1 } } { 2 }$

H) $e ^ { - 1 }$

Question 18

Use Simpson's Rule with n = 4 to approximate the integral $\int _ { 1 } ^ { 3 } \frac { 1 } { x } d x$

Accepted Answer

A)  $\frac { 33 } { 10 }$ 
B)  $\frac { 66 } { 10 }$ 
C)  $\frac { 12 } { 10 }$ 
D)  $\frac { 21 } { 20 }$ 
E)  $\frac { 11 } { 10 }$ 
F) $\frac { 22 } { 10 }$ 
G) $\frac { 6 } { 10 }$ 
H) $\frac { 21 } { 10 }$ 
A)  $\frac { 33 } { 10 }$ 
B)  $\frac { 66 } { 10 }$ 
C)  $\frac { 12 } { 10 }$ 
D)  $\frac { 21 } { 20 }$ 
E)  $\frac { 11 } { 10 }$ 
F) $\frac { 22 } { 10 }$ 
G) $\frac { 6 } { 10 }$ 
H) $\frac { 21 } { 10 }$

Question 19

Evaluate the improper integral $\int _ { 0 } ^ { 1 } x ^ { - 1 / 2 } d x$

Accepted Answer

A) 
$\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 2 }$ 
C) 1 
D) 2 
E) 
$\frac { 1 } { 4 }$ 
F)3
G)4
H)Divergent 
A) 
$\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 2 }$ 
C) 1 
D) 2 
E) 
$\frac { 1 } { 4 }$ 
F)3
G)4
H)Divergent

Question 20

Find the value of the integral $\int _ { 0 } ^ { \sqrt { 2 } / s } \frac { 1 } { \sqrt { 4 - 25 x ^ { 2 } } } d x$

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $\frac { \pi } { 5 }$ 
C)  $\frac { \pi } { 10 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { \pi } { 20 }$ 
F) $\frac { \pi } { 40 }$

G) $\frac { \pi } { 2 }$

H) $\frac { 1 } { 20 }$ 
A)  $\frac { \pi } { 4 }$ 
B)  $\frac { \pi } { 5 }$ 
C)  $\frac { \pi } { 10 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { \pi } { 20 }$ 
F) $\frac { \pi } { 40 }$

G) $\frac { \pi } { 2 }$

H) $\frac { 1 } { 20 }$

Express $\int _ { 4 } ^ { 64 } [ \sin ( \pi x ) ] d x$ as the limit of a sum.

Let f (x) = $2 ^ { 6 x }$ on the interval [0,1]. Let the interval be divided into three equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 3 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i }$ if each $x _ { i } ^ { * }$ is the midpoint of its subinterval.

Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } t ^ { 2 } d t \text {. Find the value of } f ^ { \prime } ( 1 ) \text {. }$

Let $f ( x ) = \int _ { 1 } ^ { x ^ { 2 } } \frac { 4 - u } { \sin u + 2 } d u$ Find the interval(s) on which $f ( x )$ is increasing.

Let f be a twice differentiable function such that f(0) = 5, f(3) = 1, and $f ^ { \prime }$ (3) = $- 2$ . Determine the value of $\int _ { 0 } ^ { 3 } x \cdot f ^ { \prime \prime } ( x ) d x$ .

Find the value of the integral $\int _ { 0 } ^ { x / 3 } \sec x \tan x ( 1 + \sec x ) d x$

Below is a table of values for a continuous function f. x -1 -0.5 0 0.5 1 f(x) 2.0 1.5 -1.0 0.5 -3.0 (a) Use the Trapezoidal Rule with n = 4 to approximate $\int _ { - 1 } ^ { 1 } f ( x ) d x$ (b) Use Simpson's Rule with n = 4 to approximate $\int _ { - 1 } ^ { 1 } f ( x ) d x$

If $\int _ { 0 } ^ { 4 } f ( x ) d x = 5 \text { and } \int _ { 0 } ^ { 4 } g ( x ) d x = 2 \text {, find the value of } \int _ { 0 } ^ { 4 } f ( x ) g ( x ) d x \text {. }$

Evaluate the following integrals: (a) $\int x ^ { 3 } e ^ { \left( x ^ { 4 } + 1 \right) } d x$ (b) $\int \frac { e ^ { ( \sqrt { x } ) } } { \sqrt { x } } d x$ (c) $\int \frac { e ^ { ( 1 / x ) } } { x ^ { 2 } } d x$ (d) $\int \frac { e ^ { ( \arctan x ) } } { 1 + x ^ { 2 } } d x$

If $\int _ { 3 } ^ { b } 3 x ^ { 2 } d x = 37$ find the value of b.

Use the Comparison Theorem to determine whether $\int _ { 0 } ^ { \infty } \frac { 1 } { \sqrt { x } + 1 } d x$ is convergent or divergent. Justify your answer.

Estimate $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } } d x$ using the Midpoint Rule with n = 4. Then use the error bound $\left| E _ { M } \right| \leq \frac { k ( b - a ) ^ { 3 } } { 24 n ^ { 2 } }$ to estimate the accuracy.

Let $f ( x ) = \int _ { 0 } ^ { 5 } t ^ { 4 } d t . \text { Find } f ^ { \prime } ( x )$

Suppose f (x) is a increasing function on the interval $1 \leq x \leq 4$ . If you estimate $\int _ { 1 } ^ { 4 } f ( x ) d x$ using right endpoints, explain why the estimate is an overestimate or an underestimate.

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.(a) $\int _ { 0 } ^ { 2 } \frac { 1 } { ( x - 1 ) ^ { 2 } } d x$ (b) $\int _ { 0 } ^ { 3 } \frac { d x } { ( x - 1 ) ^ { 2 / 3 } } d x$ (c) $\int _ { 0 } ^ { x } \sec ^ { 2 } t d t$

Find the value of the integral $\int _ { 0 } ^ { x / 2 } \cos ^ { 3 } x d x$

Find the value of the integral $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x ^ { 3 } } d x$

Use Simpson's Rule with n = 4 to approximate the integral $\int _ { 1 } ^ { 3 } \frac { 1 } { x } d x$

Evaluate the improper integral $\int _ { 0 } ^ { 1 } x ^ { - 1 / 2 } d x$

Find the value of the integral $\int _ { 0 } ^ { \sqrt { 2 } / s } \frac { 1 } { \sqrt { 4 - 25 x ^ { 2 } } } d x$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Exam 5: Integrals

Express ∫464[sin⁡(πx)]dx\int _ { 4 } ^ { 64 } [ \sin ( \pi x ) ] d x∫464​[sin(πx)]dx as the limit of a sum.

Let f(x)=∫0x2t2dt. Find the value of f′(1). f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } t ^ { 2 } d t \text {. Find the value of } f ^ { \prime } ( 1 ) \text {. }f(x)=∫0x2​t2dt. Find the value of f′(1).

Let f(x)=∫1x24−usin⁡u+2duf ( x ) = \int _ { 1 } ^ { x ^ { 2 } } \frac { 4 - u } { \sin u + 2 } d uf(x)=∫1x2​sinu+24−u​du Find the interval(s) on which f(x)f ( x )f(x) is increasing.

Let f be a twice differentiable function such that f(0) = 5, f(3) = 1, and f′f ^ { \prime }f′ (3) = −2- 2−2 . Determine the value of ∫03x⋅f′′(x)dx\int _ { 0 } ^ { 3 } x \cdot f ^ { \prime \prime } ( x ) d x∫03​x⋅f′′(x)dx .

Find the value of the integral ∫0x/3sec⁡xtan⁡x(1+sec⁡x)dx\int _ { 0 } ^ { x / 3 } \sec x \tan x ( 1 + \sec x ) d x∫0x/3​secxtanx(1+secx)dx

If ∫3b3x2dx=37\int _ { 3 } ^ { b } 3 x ^ { 2 } d x = 37∫3b​3x2dx=37 find the value of b.

Use the Comparison Theorem to determine whether ∫0∞1x+1dx\int _ { 0 } ^ { \infty } \frac { 1 } { \sqrt { x } + 1 } d x∫0∞​x​+11​dx is convergent or divergent. Justify your answer.

Let f(x)=∫05t4dt. Find f′(x)f ( x ) = \int _ { 0 } ^ { 5 } t ^ { 4 } d t . \text { Find } f ^ { \prime } ( x )f(x)=∫05​t4dt. Find f′(x)

Suppose f (x) is a increasing function on the interval 1≤x≤41 \leq x \leq 41≤x≤4 . If you estimate ∫14f(x)dx\int _ { 1 } ^ { 4 } f ( x ) d x∫14​f(x)dx using right endpoints, explain why the estimate is an overestimate or an underestimate.

Find the value of the integral ∫0x/2cos⁡3xdx\int _ { 0 } ^ { x / 2 } \cos ^ { 3 } x d x∫0x/2​cos3xdx

Find the value of the integral ∫01x2e−x3dx\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x ^ { 3 } } d x∫01​x2e−x3dx

Use Simpson's Rule with n = 4 to approximate the integral ∫131xdx\int _ { 1 } ^ { 3 } \frac { 1 } { x } d x∫13​x1​dx

Evaluate the improper integral ∫01x−1/2dx\int _ { 0 } ^ { 1 } x ^ { - 1 / 2 } d x∫01​x−1/2dx

Find the value of the integral ∫02/s14−25x2dx\int _ { 0 } ^ { \sqrt { 2 } / s } \frac { 1 } { \sqrt { 4 - 25 x ^ { 2 } } } d x∫02​/s​4−25x2​1​dx

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Express $\int _ { 4 } ^ { 64 } [ \sin ( \pi x ) ] d x$ as the limit of a sum.

Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } t ^ { 2 } d t \text {. Find the value of } f ^ { \prime } ( 1 ) \text {. }$

Let $f ( x ) = \int _ { 1 } ^ { x ^ { 2 } } \frac { 4 - u } { \sin u + 2 } d u$ Find the interval(s) on which $f ( x )$ is increasing.

Let f be a twice differentiable function such that f(0) = 5, f(3) = 1, and $f ^ { \prime }$ (3) = $- 2$ . Determine the value of $\int _ { 0 } ^ { 3 } x \cdot f ^ { \prime \prime } ( x ) d x$ .

Find the value of the integral $\int _ { 0 } ^ { x / 3 } \sec x \tan x ( 1 + \sec x ) d x$

If $\int _ { 3 } ^ { b } 3 x ^ { 2 } d x = 37$ find the value of b.

Use the Comparison Theorem to determine whether $\int _ { 0 } ^ { \infty } \frac { 1 } { \sqrt { x } + 1 } d x$ is convergent or divergent. Justify your answer.

Let $f ( x ) = \int _ { 0 } ^ { 5 } t ^ { 4 } d t . \text { Find } f ^ { \prime } ( x )$

Suppose f (x) is a increasing function on the interval $1 \leq x \leq 4$ . If you estimate $\int _ { 1 } ^ { 4 } f ( x ) d x$ using right endpoints, explain why the estimate is an overestimate or an underestimate.

Find the value of the integral $\int _ { 0 } ^ { x / 2 } \cos ^ { 3 } x d x$

Find the value of the integral $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x ^ { 3 } } d x$

Use Simpson's Rule with n = 4 to approximate the integral $\int _ { 1 } ^ { 3 } \frac { 1 } { x } d x$

Evaluate the improper integral $\int _ { 0 } ^ { 1 } x ^ { - 1 / 2 } d x$

Find the value of the integral $\int _ { 0 } ^ { \sqrt { 2 } / s } \frac { 1 } { \sqrt { 4 - 25 x ^ { 2 } } } d x$