Question 1

Given $f ( x ) = \sin x$ over the interval [0, 2 $\pi$ ], find: 
(a) the signed area,
 (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 2 $\pi$

Accepted Answer

The answer of Given \(f ( x ) = \sin...

Question 2

Evaluate $\int _ { - 1 } ^ { 4 } \left| 9 - x ^ { 2 } \right| d x$

Accepted Answer

A)  20 
B)  15 
C)  12 
D)  30 
A)  20 
B)  15 
C)  12 
D)  30

Question 3

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } - 2 k + 3 \right)$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Accepted Answer

The answer of Find a formula for the sum \(\sum...

Question 4

Find a formula for the sum and then use it to calculate the sum for n = 100, n = 500, and n = 1000. $$\sum _ { k = 1 } ^ { n } ( 2 - k )$$

Accepted Answer

A)  $\frac { 3 n + n ^ { 2 } } { 2 }$ 
B)  $\frac { 4 n - n ^ { 2 } } { 2 }$ 
C)  $\frac { 3 n - n ^ { 2 } } { 2 }$ 
D)  $\frac { 2 n - n ^ { 2 } } { 2 }$ 
A)  $\frac { 3 n + n ^ { 2 } } { 2 }$ 
B)  $\frac { 4 n - n ^ { 2 } } { 2 }$ 
C)  $\frac { 3 n - n ^ { 2 } } { 2 }$ 
D)  $\frac { 2 n - n ^ { 2 } } { 2 }$

Question 5

Write out each sum in expanded form, and then calculate the value of the sum. $$\sum _ { k = 1 } ^ { 6 } \ln k$$

Accepted Answer

The answer of Write out each sum in expanded form,...

Question 6

Write the following sum in sigma notation. $$\frac { 1 } { 8 } + \frac { 1 } { 27 } + \frac { 1 } { 64 } + \frac { 1 } { 125 } + \frac { 1 } { 216 }$$

Accepted Answer

A)  $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { k ^ { 2 } }$ 
B)  $\sum _ { k = 2 } ^ { 7 } \frac { 1 } { k ^ { 3 } }$ 
C)  $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { 2 k ^ { 2 } }$ 
D)  $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { k ^ { 3 } }$ 
A)  $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { k ^ { 2 } }$ 
B)  $\sum _ { k = 2 } ^ { 7 } \frac { 1 } { k ^ { 3 } }$ 
C)  $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { 2 k ^ { 2 } }$ 
D)  $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { k ^ { 3 } }$

Question 7

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \frac { \left( k ^ { 2 } + 2 k + 2 \right) } { n ^ { 3 } }$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Accepted Answer

The answer of Find a formula for the sum \(\sum...

Question 8

Evaluate $\int _ { 0 } ^ { 1 } \frac { e ^ { 3 x } - x e ^ { 2 x } } { e ^ { 2 x } } d x$

Accepted Answer

A)  $e - \frac { 1 } { 2 }$ 
B)  $e - \frac { 3 } { 2 }$ 
C)  $e + \frac { 1 } { 2 }$ 
D)  $e + \frac { 3 } { 2 }$ 
A)  $e - \frac { 1 } { 2 }$ 
B)  $e - \frac { 3 } { 2 }$ 
C)  $e + \frac { 1 } { 2 }$ 
D)  $e + \frac { 3 } { 2 }$

Question 9

Use geometry to find the exact value of $\int _ { - 2 } ^ { 6 } 10 d x$

Accepted Answer

A)  60 
B)  80 
C)  100 
D)  70 
A)  60 
B)  80 
C)  100 
D)  70

Question 10

Find $\int 2 \tan ^ { 2 } x d x$

Accepted Answer

A)  $2 \sec x - 2 x + C$ 
B)  $\frac { 2 } { 3 } \tan ^ { 3 } x + C$ 
C)  $2 \tan x + 2 x + C$ 
D)  $2 \tan x - 2 x + C$ 
A)  $2 \sec x - 2 x + C$ 
B)  $\frac { 2 } { 3 } \tan ^ { 3 } x + C$ 
C)  $2 \tan x + 2 x + C$ 
D)  $2 \tan x - 2 x + C$

Question 11

Given $f ( x ) = 3 x ^ { 2 }$ over the interval [-1, 2], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -1 to x = 2.

Accepted Answer

The answer of Given \(f ( x ) = 3...

Question 12

Find $\frac { d ^ { 2 } } { d x ^ { 2 } } \int _ { x ^ { 4 } } ^ { 2 } \ln t d t$

Accepted Answer

The answer of Find \(\frac { d ^ { 2...

Question 13

Write out each sum in expanded form, and then calculate the value of the sum. $$\sum _ { k = 2 } ^ { 7 } k ^ { 2 }$$

Accepted Answer

4 + 9 + 16

Question 14

Find $\int \frac { \left( x ^ { 3 } + 2 x \right) ^ { 2 } } { x ^ { 3 } } d x$

Accepted Answer

A)  $\frac { 1 } { 4 } x ^ { 4 } + x ^ { 2 } + 4 x + C$ 
B)  $\frac { 1 } { 4 } x ^ { 4 } + x ^ { 2 } + 4 \ln | x | + C$ 
C)  $\frac { 1 } { 4 } x ^ { 4 } + 2 x ^ { 2 } + \frac { 4 } { x ^ { 2 } } + C$ 
D)  $\frac { 1 } { 4 } x ^ { 4 } + 2 x ^ { 2 } + 4 \ln | x | + C$ 
A)  $\frac { 1 } { 4 } x ^ { 4 } + x ^ { 2 } + 4 x + C$ 
B)  $\frac { 1 } { 4 } x ^ { 4 } + x ^ { 2 } + 4 \ln | x | + C$ 
C)  $\frac { 1 } { 4 } x ^ { 4 } + 2 x ^ { 2 } + \frac { 4 } { x ^ { 2 } } + C$ 
D)  $\frac { 1 } { 4 } x ^ { 4 } + 2 x ^ { 2 } + 4 \ln | x | + C$

Question 15

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x + 2$ , $g ( x ) = 5 - x$ , and the x-axis from x = 0 to x = 6.

Accepted Answer

A)  45/4 
B)  11 
C)  43/4 
D)  45/2 
A)  45/4 
B)  11 
C)  43/4 
D)  45/2

Question 16

Evaluate $\int _ { 1 } ^ { 2 } \frac { \left( x ^ { 2 } + 2 \right) ^ { 2 } } { x ^ { 3 } } d x$

Accepted Answer

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

Question 17

If $\int _ { - 1 } ^ { 4 } f ( x ) d x = 3$ , $\int _ { - 3 } ^ { 4 } f ( x ) d x = 6$ , $\int _ { - 1 }^4 g ( x ) d x = - 2$ and $\int _ { 4 } ^ { 6 } g ( x ) d x = 5$ , find the following, if possible. 
(a) $\int _ { - 3 } ^ { - 1 } f ( x ) d x$ 
(b) $\int _ { - 1 }^6 ( g ( x ) + 3 ) d x$ 
(c) $\int _ { - 1 } ^ { 4 } ( 2 f ( x ) + x ) d x$

Accepted Answer

(a) 3 (b)

Question 18

Evaluate $\int _ { 1 } ^ { 3 } \frac { 2 } { x } d x$

Accepted Answer

A)  ln6 
B)  ln3 
C)  ln9 
D)  1 
A)  ln6 
B)  ln3 
C)  ln9 
D)  1

Question 19

Find $\int \left( 6 x ^ { 2 } + 5 - 2 e ^ { x } \right) d x$

Accepted Answer

A)  $2 x ^ { 3 } + 5 - 2 e ^ { x } + C$ 
B)  $x ^ { 3 } + 5 x - 2 e ^ { x } + C$ 
C)  $2 x ^ { 3 } + 5 x - 2 e ^ { x } + C$ 
D)  $2 x ^ { 3 } + 5 x + 2 e ^ { x } + C$ 
A)  $2 x ^ { 3 } + 5 - 2 e ^ { x } + C$ 
B)  $x ^ { 3 } + 5 x - 2 e ^ { x } + C$ 
C)  $2 x ^ { 3 } + 5 x - 2 e ^ { x } + C$ 
D)  $2 x ^ { 3 } + 5 x + 2 e ^ { x } + C$

Question 20

Approximate the area between the graph of $f ( x ) = x ^ { 2 } + 4 x$ and the x-axis on the interval [2, 4], using a left sum with (a) n = 4, (b) n = 6.

Accepted Answer

(a) 302/8

Given $f ( x ) = \sin x$ over the interval [0, 2 $\pi$ ], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 2 $\pi$

Evaluate $\int _ { - 1 } ^ { 4 } \left| 9 - x ^ { 2 } \right| d x$

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } - 2 k + 3 \right)$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Find a formula for the sum and then use it to calculate the sum for n = 100, n = 500, and n = 1000. $\sum _ { k = 1 } ^ { n } ( 2 - k )$

Write out each sum in expanded form, and then calculate the value of the sum. $\sum _ { k = 1 } ^ { 6 } \ln k$

Write the following sum in sigma notation. $\frac { 1 } { 8 } + \frac { 1 } { 27 } + \frac { 1 } { 64 } + \frac { 1 } { 125 } + \frac { 1 } { 216 }$

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \frac { \left( k ^ { 2 } + 2 k + 2 \right) } { n ^ { 3 } }$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Evaluate $\int _ { 0 } ^ { 1 } \frac { e ^ { 3 x } - x e ^ { 2 x } } { e ^ { 2 x } } d x$

Use geometry to find the exact value of $\int _ { - 2 } ^ { 6 } 10 d x$

Find $\int 2 \tan ^ { 2 } x d x$

Given $f ( x ) = 3 x ^ { 2 }$ over the interval [-1, 2], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -1 to x = 2.

Find $\frac { d ^ { 2 } } { d x ^ { 2 } } \int _ { x ^ { 4 } } ^ { 2 } \ln t d t$

Write out each sum in expanded form, and then calculate the value of the sum. $\sum _ { k = 2 } ^ { 7 } k ^ { 2 }$

Find $\int \frac { \left( x ^ { 3 } + 2 x \right) ^ { 2 } } { x ^ { 3 } } d x$

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x + 2$ , $g ( x ) = 5 - x$ , and the x-axis from x = 0 to x = 6.

Evaluate $\int _ { 1 } ^ { 2 } \frac { \left( x ^ { 2 } + 2 \right) ^ { 2 } } { x ^ { 3 } } d x$

Evaluate $\int _ { 1 } ^ { 3 } \frac { 2 } { x } d x$

Find $\int \left( 6 x ^ { 2 } + 5 - 2 e ^ { x } \right) d x$

Approximate the area between the graph of $f ( x ) = x ^ { 2 } + 4 x$ and the x-axis on the interval [2, 4], using a left sum with (a) n = 4, (b) n = 6.

Limits

Derivatives

Applications of the Derivative

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 4: Definite Integrals

Given f(x)=sin⁡xf ( x ) = \sin xf(x)=sinx over the interval [0, 2 π\piπ ], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 2 π\piπ

Evaluate ∫−14∣9−x2∣dx\int _ { - 1 } ^ { 4 } \left| 9 - x ^ { 2 } \right| d x∫−14​​9−x2​dx

Find a formula for the sum ∑k=1n(k2−2k+3)\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } - 2 k + 3 \right)∑k=1n​(k2−2k+3) and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Find a formula for the sum and then use it to calculate the sum for n = 100, n = 500, and n = 1000. ∑k=1n(2−k)\sum _ { k = 1 } ^ { n } ( 2 - k )∑k=1n​(2−k)

Write out each sum in expanded form, and then calculate the value of the sum. ∑k=16ln⁡k\sum _ { k = 1 } ^ { 6 } \ln k∑k=16​lnk

Write the following sum in sigma notation. 18+127+164+1125+1216\frac { 1 } { 8 } + \frac { 1 } { 27 } + \frac { 1 } { 64 } + \frac { 1 } { 125 } + \frac { 1 } { 216 }81​+271​+641​+1251​+2161​

Find a formula for the sum ∑k=1n(k2+2k+2)n3\sum _ { k = 1 } ^ { n } \frac { \left( k ^ { 2 } + 2 k + 2 \right) } { n ^ { 3 } }∑k=1n​n3(k2+2k+2)​ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Evaluate ∫01e3x−xe2xe2xdx\int _ { 0 } ^ { 1 } \frac { e ^ { 3 x } - x e ^ { 2 x } } { e ^ { 2 x } } d x∫01​e2xe3x−xe2x​dx

Use geometry to find the exact value of ∫−2610dx\int _ { - 2 } ^ { 6 } 10 d x∫−26​10dx

Find ∫2tan⁡2xdx\int 2 \tan ^ { 2 } x d x∫2tan2xdx

Given f(x)=3x2f ( x ) = 3 x ^ { 2 }f(x)=3x2 over the interval [-1, 2], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -1 to x = 2.

Find d2dx2∫x42ln⁡tdt\frac { d ^ { 2 } } { d x ^ { 2 } } \int _ { x ^ { 4 } } ^ { 2 } \ln t d tdx2d2​∫x42​lntdt

Write out each sum in expanded form, and then calculate the value of the sum. ∑k=27k2\sum _ { k = 2 } ^ { 7 } k ^ { 2 }∑k=27​k2

Find ∫(x3+2x)2x3dx\int \frac { \left( x ^ { 3 } + 2 x \right) ^ { 2 } } { x ^ { 3 } } d x∫x3(x3+2x)2​dx

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of f(x)=x+2f ( x ) = x + 2f(x)=x+2 , g(x)=5−xg ( x ) = 5 - xg(x)=5−x , and the x-axis from x = 0 to x = 6.

Evaluate ∫12(x2+2)2x3dx\int _ { 1 } ^ { 2 } \frac { \left( x ^ { 2 } + 2 \right) ^ { 2 } } { x ^ { 3 } } d x∫12​x3(x2+2)2​dx

Evaluate ∫132xdx\int _ { 1 } ^ { 3 } \frac { 2 } { x } d x∫13​x2​dx

Find ∫(6x2+5−2ex)dx\int \left( 6 x ^ { 2 } + 5 - 2 e ^ { x } \right) d x∫(6x2+5−2ex)dx

Approximate the area between the graph of f(x)=x2+4xf ( x ) = x ^ { 2 } + 4 xf(x)=x2+4x and the x-axis on the interval [2, 4], using a left sum with (a) n = 4, (b) n = 6.

Limits

Derivatives

Applications of the Derivative

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Given $f ( x ) = \sin x$ over the interval [0, 2 $\pi$ ], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 2 $\pi$

Evaluate $\int _ { - 1 } ^ { 4 } \left| 9 - x ^ { 2 } \right| d x$

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } - 2 k + 3 \right)$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Find a formula for the sum and then use it to calculate the sum for n = 100, n = 500, and n = 1000. $\sum _ { k = 1 } ^ { n } ( 2 - k )$

Write out each sum in expanded form, and then calculate the value of the sum. $\sum _ { k = 1 } ^ { 6 } \ln k$

Write the following sum in sigma notation. $\frac { 1 } { 8 } + \frac { 1 } { 27 } + \frac { 1 } { 64 } + \frac { 1 } { 125 } + \frac { 1 } { 216 }$

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \frac { \left( k ^ { 2 } + 2 k + 2 \right) } { n ^ { 3 } }$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Evaluate $\int _ { 0 } ^ { 1 } \frac { e ^ { 3 x } - x e ^ { 2 x } } { e ^ { 2 x } } d x$

Use geometry to find the exact value of $\int _ { - 2 } ^ { 6 } 10 d x$

Find $\int 2 \tan ^ { 2 } x d x$

Given $f ( x ) = 3 x ^ { 2 }$ over the interval [-1, 2], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -1 to x = 2.

Find $\frac { d ^ { 2 } } { d x ^ { 2 } } \int _ { x ^ { 4 } } ^ { 2 } \ln t d t$

Write out each sum in expanded form, and then calculate the value of the sum. $\sum _ { k = 2 } ^ { 7 } k ^ { 2 }$

Find $\int \frac { \left( x ^ { 3 } + 2 x \right) ^ { 2 } } { x ^ { 3 } } d x$

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x + 2$ , $g ( x ) = 5 - x$ , and the x-axis from x = 0 to x = 6.

Evaluate $\int _ { 1 } ^ { 2 } \frac { \left( x ^ { 2 } + 2 \right) ^ { 2 } } { x ^ { 3 } } d x$

Evaluate $\int _ { 1 } ^ { 3 } \frac { 2 } { x } d x$

Find $\int \left( 6 x ^ { 2 } + 5 - 2 e ^ { x } \right) d x$

Approximate the area between the graph of $f ( x ) = x ^ { 2 } + 4 x$ and the x-axis on the interval [2, 4], using a left sum with (a) n = 4, (b) n = 6.