Question 1

Use geometry to find the exact value of $\int _ { 0 } ( 3 - x ) d x$

Accepted Answer

A)  5 
B)  3 
C)  6 
D)  4 
A)  5 
B)  3 
C)  6 
D)  4 D

Question 2

Use geometry to find the exact value of $\int _ { - 2 } ^ { 2 } \sqrt { 4 - x ^ { 2 } } d x$

Accepted Answer

A)  $\pi$ 
B)  $\pi$ /2 
C)  4 $\pi$ 
D)  2 $\pi$ 
A)  $\pi$ 
B)  $\pi$ /2 
C)  4 $\pi$ 
D)  2 $\pi$ D

Question 3

Evaluate $\int _ { 0 } ^ { \pi / 2 } ( 2 \sin x - 3 \cos x ) d x$

Accepted Answer

A)  1 
B)  0 
C)  -1 
D)  $1 / 2$ 
A)  1 
B)  0 
C)  -1 
D)  $1 / 2$ C

Question 4

Find the exact average value of $f ( x ) = 2 e ^ { x }$ from x = -1 to x = 3.

Accepted Answer

The answer of Find the exact average value of \(f...

Question 5

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

Accepted Answer

A)  $\ln | x | - \frac { 2 } { x } - \frac { 1 } { x ^ { 4 } } + C$ 
B)  $- \frac { 2 } { x ^ { 2 } } - \frac { 1 } { x ^ { 4 } } - \frac { 20 } { x ^ { 6 } } + C$ 
C)  $\ln | x | - \frac { 2 } { x ^ { 2 } } - \frac { 1 } { x ^ { 4 } } + C$ 
D)  $\ln | x | + \frac { 2 } { x ^ { 2 } } + \frac { 1 } { x ^ { 4 } } + C$ 
A)  $\ln | x | - \frac { 2 } { x } - \frac { 1 } { x ^ { 4 } } + C$ 
B)  $- \frac { 2 } { x ^ { 2 } } - \frac { 1 } { x ^ { 4 } } - \frac { 20 } { x ^ { 6 } } + C$ 
C)  $\ln | x | - \frac { 2 } { x ^ { 2 } } - \frac { 1 } { x ^ { 4 } } + C$ 
D)  $\ln | x | + \frac { 2 } { x ^ { 2 } } + \frac { 1 } { x ^ { 4 } } + C$

Question 6

Write the following sum in sigma notation. $$2 + \frac { 3 } { 4 } + \frac { 4 } { 9 } + \frac { 5 } { 16 } + \frac { 6 } { 25 } + \frac { 7 } { 36 } + \frac { 8 } { 49 }$$

Accepted Answer

A)  $\sum _ { k = 3 } ^ { 9 } \frac { k + 1 } { k }$ 
B)  $\sum _ { k = 1 } ^ { 7 } \frac { k + 1 } { k }$ 
C)  $\sum _ { k = 1 } ^ { 7 } \frac { k + 1 } { k ^ { 2 } }$ 
D)  $\sum _ { k = 3 } ^ { 10 } \frac { k + 1 } { k ^ { 2 } }$ 
A)  $\sum _ { k = 3 } ^ { 9 } \frac { k + 1 } { k }$ 
B)  $\sum _ { k = 1 } ^ { 7 } \frac { k + 1 } { k }$ 
C)  $\sum _ { k = 1 } ^ { 7 } \frac { k + 1 } { k ^ { 2 } }$ 
D)  $\sum _ { k = 3 } ^ { 10 } \frac { k + 1 } { k ^ { 2 } }$

Question 7

Given $f ( x ) = 1 - e ^ { x }$ 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 ) = 1...

Question 8

Evaluate $\int _ { 1 } 5 \left( 3 ^ { x } \right) d x$

Accepted Answer

A)  $\frac { 100 } { \ln 3 }$ 
B)  $120 \ln 3$ 
C)  $100 \ln 3$ 
D)  $\frac { 120 } { \ln 3 }$ 
A)  $\frac { 100 } { \ln 3 }$ 
B)  $120 \ln 3$ 
C)  $100 \ln 3$ 
D)  $\frac { 120 } { \ln 3 }$

Question 9

Approximate the area between the graph of $f ( x ) = ( x - 2 ) ^ { 2 } + 1$ and the x-axis on the interval [1, 5], using a right sum with (a) n = 2, (b) n = 4.

Accepted Answer

The answer of Approximate the area between the graph of...

Question 10

Given $f ( x ) = x ^ { 2 } + x - 6$ over the interval [-3, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -3 to x = 4.

Accepted Answer

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

Question 11

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

Accepted Answer

A)  $2 e ^ { 4 } \left( e ^ { 2 } + 1 \right)$ 
B)  $2 e ^ { 3 } \left( e ^ { 2 } + 1 \right)$ 
C)  $2 e ^ { 3 } \left( e ^ { 2 } - 1 \right)$ 
D)  $2 e ^ { 4 } \left( e ^ { 2 } - 1 \right)$ 
A)  $2 e ^ { 4 } \left( e ^ { 2 } + 1 \right)$ 
B)  $2 e ^ { 3 } \left( e ^ { 2 } + 1 \right)$ 
C)  $2 e ^ { 3 } \left( e ^ { 2 } - 1 \right)$ 
D)  $2 e ^ { 4 } \left( e ^ { 2 } - 1 \right)$

Question 12

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

Accepted Answer

(a) 382/8

Question 13

Evaluate $\int _ { 0 } ^ { 1 / 2 } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$

Accepted Answer

A)  $\pi$ /3 
B)  $\pi$ /4 
C)  $\pi$ 
D)  $\pi$ /6 
A)  $\pi$ /3 
B)  $\pi$ /4 
C)  $\pi$ 
D)  $\pi$ /6

Question 14

Evaluate $\int _ { - 1 } \left| e ^ { 3 x } - 1 \right| d x$

Accepted Answer

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

Question 15

Given $f ( x ) = ( x + 1 ) ^ { 2 } - 2$ , find a number c on the interval (-3, 0) such that f(c) is the average value of f on [-3, 0].

Accepted Answer

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

Question 16

Find $\frac { d ^ { 2 } } { d x ^ { 2 } } \int _ { 1 } ^ { x } \frac { \cos t } { 2 t } d t$

Accepted Answer

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

Question 17

Approximate the area between the graph of $f ( x ) = 9 - x ^ { 2 }$ and the x-axis on the interval [0, 3], for n = 6, using (a) left sums (b) right sums.

Accepted Answer

The answer of Approximate the area between the graph of...

Question 18

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = \sin x$ and $g ( x ) = \cos x$ from x = 0 to x = $\pi$ /2.

Accepted Answer

A)  $\sqrt { 2 } - 2$ 
B)  $2 \sqrt { 2 } - 2$ 
C)  $\sqrt { 2 } + 2$ 
D)  $2 \sqrt { 2 } + 2$ 
A)  $\sqrt { 2 } - 2$ 
B)  $2 \sqrt { 2 } - 2$ 
C)  $\sqrt { 2 } + 2$ 
D)  $2 \sqrt { 2 } + 2$

Question 19

Find the exact average value of $f ( x ) = 9 - x ^ { 2 }$ from x = -3 to x = 3.

Accepted Answer

A)  0 
B)  36 
C)  18 
D)  6 
A)  0 
B)  36 
C)  18 
D)  6

Question 20

Find $\frac { d } { d x } \int _ { 2 } ^ { \tan x } 3 t ^ { 3 } d t$

Accepted Answer

The answer of Find \(\frac { d } { d...

Use geometry to find the exact value of $\int _ { 0 } ( 3 - x ) d x$

Use geometry to find the exact value of $\int _ { - 2 } ^ { 2 } \sqrt { 4 - x ^ { 2 } } d x$

Evaluate $\int _ { 0 } ^ { \pi / 2 } ( 2 \sin x - 3 \cos x ) d x$

Find the exact average value of $f ( x ) = 2 e ^ { x }$ from x = -1 to x = 3.

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

Write the following sum in sigma notation. $2 + \frac { 3 } { 4 } + \frac { 4 } { 9 } + \frac { 5 } { 16 } + \frac { 6 } { 25 } + \frac { 7 } { 36 } + \frac { 8 } { 49 }$

Given $f ( x ) = 1 - e ^ { x }$ 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.

Evaluate $\int _ { 1 } 5 \left( 3 ^ { x } \right) d x$

Approximate the area between the graph of $f ( x ) = ( x - 2 ) ^ { 2 } + 1$ and the x-axis on the interval [1, 5], using a right sum with (a) n = 2, (b) n = 4.

Given $f ( x ) = x ^ { 2 } + x - 6$ over the interval [-3, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -3 to x = 4.

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

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

Evaluate $\int _ { 0 } ^ { 1 / 2 } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$

Evaluate $\int _ { - 1 } \left| e ^ { 3 x } - 1 \right| d x$

Given $f ( x ) = ( x + 1 ) ^ { 2 } - 2$ , find a number c on the interval (-3, 0) such that f(c) is the average value of f on [-3, 0].

Find $\frac { d ^ { 2 } } { d x ^ { 2 } } \int _ { 1 } ^ { x } \frac { \cos t } { 2 t } d t$

Approximate the area between the graph of $f ( x ) = 9 - x ^ { 2 }$ and the x-axis on the interval [0, 3], for n = 6, using (a) left sums (b) right sums.

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = \sin x$ and $g ( x ) = \cos x$ from x = 0 to x = $\pi$ /2.

Find the exact average value of $f ( x ) = 9 - x ^ { 2 }$ from x = -3 to x = 3.

Find $\frac { d } { d x } \int _ { 2 } ^ { \tan x } 3 t ^ { 3 } d t$

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Exam 4: Definite Integrals

Use geometry to find the exact value of ∫0(3−x)dx\int _ { 0 } ( 3 - x ) d x∫0​(3−x)dx

Use geometry to find the exact value of ∫−224−x2dx\int _ { - 2 } ^ { 2 } \sqrt { 4 - x ^ { 2 } } d x∫−22​4−x2​dx

Evaluate ∫0π/2(2sin⁡x−3cos⁡x)dx\int _ { 0 } ^ { \pi / 2 } ( 2 \sin x - 3 \cos x ) d x∫0π/2​(2sinx−3cosx)dx

Find the exact average value of f(x)=2exf ( x ) = 2 e ^ { x }f(x)=2ex from x = -1 to x = 3.

Find ∫(x2+2)2x5dx\int \frac { \left( x ^ { 2 } + 2 \right) ^ { 2 } } { x ^ { 5 } } d x∫x5(x2+2)2​dx

Write the following sum in sigma notation. 2+34+49+516+625+736+8492 + \frac { 3 } { 4 } + \frac { 4 } { 9 } + \frac { 5 } { 16 } + \frac { 6 } { 25 } + \frac { 7 } { 36 } + \frac { 8 } { 49 }2+43​+94​+165​+256​+367​+498​

Given f(x)=1−exf ( x ) = 1 - e ^ { x }f(x)=1−ex 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.

Evaluate ∫15(3x)dx\int _ { 1 } 5 \left( 3 ^ { x } \right) d x∫1​5(3x)dx

Approximate the area between the graph of f(x)=(x−2)2+1f ( x ) = ( x - 2 ) ^ { 2 } + 1f(x)=(x−2)2+1 and the x-axis on the interval [1, 5], using a right sum with (a) n = 2, (b) n = 4.

Given f(x)=x2+x−6f ( x ) = x ^ { 2 } + x - 6f(x)=x2+x−6 over the interval [-3, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -3 to x = 4.

Evaluate ∫12ex+3dx\int _ { 1 } 2 e ^ { x + 3 } d x∫1​2ex+3dx

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 right sum with (a) n = 4, (b) n = 6.

Evaluate ∫01/211−x2dx\int _ { 0 } ^ { 1 / 2 } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x∫01/2​1−x2​1​dx

Evaluate ∫−1∣e3x−1∣dx\int _ { - 1 } \left| e ^ { 3 x } - 1 \right| d x∫−1​​e3x−1​dx

Given f(x)=(x+1)2−2f ( x ) = ( x + 1 ) ^ { 2 } - 2f(x)=(x+1)2−2 , find a number c on the interval (-3, 0) such that f(c) is the average value of f on [-3, 0].

Find d2dx2∫1xcos⁡t2tdt\frac { d ^ { 2 } } { d x ^ { 2 } } \int _ { 1 } ^ { x } \frac { \cos t } { 2 t } d tdx2d2​∫1x​2tcost​dt

Approximate the area between the graph of f(x)=9−x2f ( x ) = 9 - x ^ { 2 }f(x)=9−x2 and the x-axis on the interval [0, 3], for n = 6, using (a) left sums (b) right sums.

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of f(x)=sin⁡xf ( x ) = \sin xf(x)=sinx and g(x)=cos⁡xg ( x ) = \cos xg(x)=cosx from x = 0 to x = π\piπ /2.

Find the exact average value of f(x)=9−x2f ( x ) = 9 - x ^ { 2 }f(x)=9−x2 from x = -3 to x = 3.

Find ddx∫2tan⁡x3t3dt\frac { d } { d x } \int _ { 2 } ^ { \tan x } 3 t ^ { 3 } d tdxd​∫2tanx​3t3dt

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

Use geometry to find the exact value of $\int _ { 0 } ( 3 - x ) d x$

Use geometry to find the exact value of $\int _ { - 2 } ^ { 2 } \sqrt { 4 - x ^ { 2 } } d x$

Evaluate $\int _ { 0 } ^ { \pi / 2 } ( 2 \sin x - 3 \cos x ) d x$

Find the exact average value of $f ( x ) = 2 e ^ { x }$ from x = -1 to x = 3.

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

Write the following sum in sigma notation. $2 + \frac { 3 } { 4 } + \frac { 4 } { 9 } + \frac { 5 } { 16 } + \frac { 6 } { 25 } + \frac { 7 } { 36 } + \frac { 8 } { 49 }$

Given $f ( x ) = 1 - e ^ { x }$ 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.

Evaluate $\int _ { 1 } 5 \left( 3 ^ { x } \right) d x$

Approximate the area between the graph of $f ( x ) = ( x - 2 ) ^ { 2 } + 1$ and the x-axis on the interval [1, 5], using a right sum with (a) n = 2, (b) n = 4.

Given $f ( x ) = x ^ { 2 } + x - 6$ over the interval [-3, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -3 to x = 4.

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

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

Evaluate $\int _ { 0 } ^ { 1 / 2 } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$

Evaluate $\int _ { - 1 } \left| e ^ { 3 x } - 1 \right| d x$

Given $f ( x ) = ( x + 1 ) ^ { 2 } - 2$ , find a number c on the interval (-3, 0) such that f(c) is the average value of f on [-3, 0].

Find $\frac { d ^ { 2 } } { d x ^ { 2 } } \int _ { 1 } ^ { x } \frac { \cos t } { 2 t } d t$

Approximate the area between the graph of $f ( x ) = 9 - x ^ { 2 }$ and the x-axis on the interval [0, 3], for n = 6, using (a) left sums (b) right sums.

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = \sin x$ and $g ( x ) = \cos x$ from x = 0 to x = $\pi$ /2.

Find the exact average value of $f ( x ) = 9 - x ^ { 2 }$ from x = -3 to x = 3.

Find $\frac { d } { d x } \int _ { 2 } ^ { \tan x } 3 t ^ { 3 } d t$