Exam 4: Applications of Differentiation

The acceleration function of a body moving along a coordinate line is $a ( t ) = - 7 \cos 2 t - 8 \sin 2 t \quad t \geq 0$ Find its velocity and position functions at any time $t$ if it is located at the origin and has an initial velocity of $4 \mathrm {~m} / \mathrm { sec }$ .

(Short Answer)

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Question 21

Express the limit as a definite integral on the given interval. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , \quad [ 6,12 ]$

(Short Answer)

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Question 22

Express the integral as a limit of sums. Then evaluate the limit. $\int _ { 0 } ^ { x } \sin 3 x d x$

(Short Answer)

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Question 23

Estimate the area from 0 to 5 under the graph of $f ( x ) = 25 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

(Short Answer)

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Question 24

If $h ^ { \prime }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 5 and 7 ?

(Short Answer)

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Question 25

Find the integral. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

(Short Answer)

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Question 26

Evaluate the integral if it exists. $\int _ { 0 } ^ { 1 } 3 x ^ { 2 } \cos \left( x ^ { 2 } \right) d x$

(Short Answer)

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Question 27

Evaluate the definite integral. $\int _ { 0 } ^ { x / 6 } \sin 8 t d t$

(Multiple Choice)

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Question 28

Find the indefinite integral. $\int x ^ { 3 } \sec ^ { 2 } x ^ { 4 } d x$

(Short Answer)

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Question 29

$\text { If } \int _ { 0 } ^ { 6 } f ( x ) d x = 12 \text { and } \int _ { 0 } ^ { 4 } f ( x ) d x = 6 \text {, find } \int _ { 4 } ^ { 6 } f ( x ) d x \text {. }$

(Short Answer)

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Question 30

Evaluate $\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d t$

(Short Answer)

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Question 31

Evaluate the indefinite integral. $\int \frac { e ^ { x } } { e ^ { x } + 5 } d x$

(Short Answer)

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Question 32

Evaluate by interpreting it in terms of areas. $\int _ { - 1 } ^ { 3 } ( 2 - x ) d x$

(Short Answer)

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Question 33

Evaluate by interpreting it in terms of areas. $\int _ { - 4 } ^ { 4 } \sqrt { 16 - x ^ { 2 } } d x$

(Short Answer)

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Question 34

Find the indefinite integral. $\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x$

(Short Answer)

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Question 35

$\text { Evaluate } \int _ { - 1 / 2 } ^ { 1 / 2 } \frac { x ^ { 7 } - 2 x ^ { 5 } + 3 x ^ { 3 } + 4 x ^ { 2 } + x - 4 } { x ^ { 2 } - 1 } d x$

(Short Answer)

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Question 36

The graph of a function $f$ on the interval $[ 0,8 ]$ is shown in the figure. Compute the Riemann sum for $f$ on $[ 0,8 ]$ using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

(Short Answer)

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Question 37

Find the indefinite integral. $\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x$

(Multiple Choice)

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Question 38

Find the indefinite integral. $\int x \sqrt { x - 8 } d x$

(Short Answer)

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Question 39

Use the following property of the definite integral to estimate the definite integral $\int _ { z / 6 } ^ { x / 4 } 4 \sin x d x$ : If $m \leq f ( x ) \leq M$ on $[ a , b ]$ , then $m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$

(Short Answer)

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Question 40

The acceleration function of a body moving along a coordinate line is $a ( t ) = - 7 \cos 2 t - 8 \sin 2 t \quad t \geq 0$ Find its velocity and position functions at any time $t$ if it is located at the origin and has an initial velocity of $4 \mathrm {~m} / \mathrm { sec }$ .

Express the limit as a definite integral on the given interval. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , \quad [ 6,12 ]$

Express the integral as a limit of sums. Then evaluate the limit. $\int _ { 0 } ^ { x } \sin 3 x d x$

Estimate the area from 0 to 5 under the graph of $f ( x ) = 25 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

If $h ^ { \prime }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 5 and 7 ?

Find the integral. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Evaluate the integral if it exists. $\int _ { 0 } ^ { 1 } 3 x ^ { 2 } \cos \left( x ^ { 2 } \right) d x$

Evaluate the definite integral. $\int _ { 0 } ^ { x / 6 } \sin 8 t d t$

Find the indefinite integral. $\int x ^ { 3 } \sec ^ { 2 } x ^ { 4 } d x$

$\text { If } \int _ { 0 } ^ { 6 } f ( x ) d x = 12 \text { and } \int _ { 0 } ^ { 4 } f ( x ) d x = 6 \text {, find } \int _ { 4 } ^ { 6 } f ( x ) d x \text {. }$

Evaluate $\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d t$

Evaluate the indefinite integral. $\int \frac { e ^ { x } } { e ^ { x } + 5 } d x$

Evaluate by interpreting it in terms of areas. $\int _ { - 1 } ^ { 3 } ( 2 - x ) d x$

Evaluate by interpreting it in terms of areas. $\int _ { - 4 } ^ { 4 } \sqrt { 16 - x ^ { 2 } } d x$

Find the indefinite integral. $\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x$

$\text { Evaluate } \int _ { - 1 / 2 } ^ { 1 / 2 } \frac { x ^ { 7 } - 2 x ^ { 5 } + 3 x ^ { 3 } + 4 x ^ { 2 } + x - 4 } { x ^ { 2 } - 1 } d x$

The graph of a function $f$ on the interval $[ 0,8 ]$ is shown in the figure. Compute the Riemann sum for $f$ on $[ 0,8 ]$ using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

Find the indefinite integral. $\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x$

Find the indefinite integral. $\int x \sqrt { x - 8 } d x$

Use the following property of the definite integral to estimate the definite integral $\int _ { z / 6 } ^ { x / 4 } 4 \sin x d x$ : If $m \leq f ( x ) \leq M$ on $[ a , b ]$ , then $m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

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Partial Derivatives

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Exam 4: Applications of Differentiation

Express the limit as a definite integral on the given interval. lim⁡n→∞∑i=1n(6risin⁡ri)Δr,[6,12]\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , \quad [ 6,12 ]limn→∞​∑i=1n​(6ri​sinri​)Δr,[6,12]

Express the integral as a limit of sums. Then evaluate the limit. ∫0xsin⁡3xdx\int _ { 0 } ^ { x } \sin 3 x d x∫0x​sin3xdx

Estimate the area from 0 to 5 under the graph of f(x)=25−x2f ( x ) = 25 - x ^ { 2 }f(x)=25−x2 using five approximating rectangles and right endpoints.

If h′h ^ { \prime }h′ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 5 and 7 ?

Find the integral. ∫tan⁡2xsec⁡6xdx\int \tan ^ { 2 } x \sec ^ { 6 } x d x∫tan2xsec6xdx

Evaluate the integral if it exists. ∫013x2cos⁡(x2)dx\int _ { 0 } ^ { 1 } 3 x ^ { 2 } \cos \left( x ^ { 2 } \right) d x∫01​3x2cos(x2)dx

Evaluate the definite integral. ∫0x/6sin⁡8tdt\int _ { 0 } ^ { x / 6 } \sin 8 t d t∫0x/6​sin8tdt

Find the indefinite integral. ∫x3sec⁡2x4dx\int x ^ { 3 } \sec ^ { 2 } x ^ { 4 } d x∫x3sec2x4dx

If ∫06f(x)dx=12 and ∫04f(x)dx=6, find ∫46f(x)dx. \text { If } \int _ { 0 } ^ { 6 } f ( x ) d x = 12 \text { and } \int _ { 0 } ^ { 4 } f ( x ) d x = 6 \text {, find } \int _ { 4 } ^ { 6 } f ( x ) d x \text {. } If ∫06​f(x)dx=12 and ∫04​f(x)dx=6, find ∫46​f(x)dx.

Evaluate ddx∫0x(earcsin⁡t)dt\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d tdxd​∫0x​(earcsint)dt

Evaluate the indefinite integral. ∫exex+5dx\int \frac { e ^ { x } } { e ^ { x } + 5 } d x∫ex+5ex​dx

Evaluate by interpreting it in terms of areas. ∫−13(2−x)dx\int _ { - 1 } ^ { 3 } ( 2 - x ) d x∫−13​(2−x)dx

Evaluate by interpreting it in terms of areas. ∫−4416−x2dx\int _ { - 4 } ^ { 4 } \sqrt { 16 - x ^ { 2 } } d x∫−44​16−x2​dx

Find the indefinite integral. ∫(5−8x3+3x6)dx\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x∫(5−8x3+3x6)dx

Evaluate ∫−1/21/2x7−2x5+3x3+4x2+x−4x2−1dx\text { Evaluate } \int _ { - 1 / 2 } ^ { 1 / 2 } \frac { x ^ { 7 } - 2 x ^ { 5 } + 3 x ^ { 3 } + 4 x ^ { 2 } + x - 4 } { x ^ { 2 } - 1 } d x Evaluate ∫−1/21/2​x2−1x7−2x5+3x3+4x2+x−4​dx

Find the indefinite integral. ∫2x9−6x2+3x9dx\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x∫x92x9−6x2+3​dx

Find the indefinite integral. ∫xx−8dx\int x \sqrt { x - 8 } d x∫xx−8​dx

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Express the limit as a definite integral on the given interval. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , \quad [ 6,12 ]$

Express the integral as a limit of sums. Then evaluate the limit. $\int _ { 0 } ^ { x } \sin 3 x d x$

Estimate the area from 0 to 5 under the graph of $f ( x ) = 25 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

If $h ^ { \prime }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 5 and 7 ?

Find the integral. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Evaluate the integral if it exists. $\int _ { 0 } ^ { 1 } 3 x ^ { 2 } \cos \left( x ^ { 2 } \right) d x$

Evaluate the definite integral. $\int _ { 0 } ^ { x / 6 } \sin 8 t d t$

Find the indefinite integral. $\int x ^ { 3 } \sec ^ { 2 } x ^ { 4 } d x$

$\text { If } \int _ { 0 } ^ { 6 } f ( x ) d x = 12 \text { and } \int _ { 0 } ^ { 4 } f ( x ) d x = 6 \text {, find } \int _ { 4 } ^ { 6 } f ( x ) d x \text {. }$

Evaluate $\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d t$

Evaluate the indefinite integral. $\int \frac { e ^ { x } } { e ^ { x } + 5 } d x$

Evaluate by interpreting it in terms of areas. $\int _ { - 1 } ^ { 3 } ( 2 - x ) d x$

Evaluate by interpreting it in terms of areas. $\int _ { - 4 } ^ { 4 } \sqrt { 16 - x ^ { 2 } } d x$

Find the indefinite integral. $\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x$

$\text { Evaluate } \int _ { - 1 / 2 } ^ { 1 / 2 } \frac { x ^ { 7 } - 2 x ^ { 5 } + 3 x ^ { 3 } + 4 x ^ { 2 } + x - 4 } { x ^ { 2 } - 1 } d x$

Find the indefinite integral. $\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x$

Find the indefinite integral. $\int x \sqrt { x - 8 } d x$