Exam 6: Applications of Integration

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln x , y = 1 , y = 5 , x = 0 ;$ about the $y -$ axis

(Multiple Choice)

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Question 61

$\text { Find } y ^ {t } \text { if } \ln x y = 2 y \sin x \text {. }$

(Short Answer)

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Question 62

Find the points of intersection of the graphs of the functions.Express your answers accurate to five decimal places. $f ( x ) = 0.2 x ^ { 2 } - 1.8 x - 3.8 ; g ( x ) = - 0.1 x ^ { 2 } + 0.8 x + 6.7$

(Short Answer)

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Question 63

Use the properties of logarithms to expand the quantity. $\log _ { 2 } \left( \frac { x ^ { 10 } y } { z ^ { 5 } } \right)$

(Short Answer)

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Question 64

Find $\left( f ^ { - 1 } \right) ^ { t } ( a )$ $f ( x ) = 7 + x ^ { 2 } + \tan \left( \frac { \pi x } { 2 } \right) , - 1 < x < 1 , a = 7$

(Short Answer)

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Question 65

Differentiate the function. $y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)$

(Multiple Choice)

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Question 66

Use logarithmic differentiation to find the derivative of the function. $y = x ^ { 6 x }$

(Short Answer)

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Question 67

$\text { Find } y ^ {t } \text { if } \ln x y = 2 y \sin x \text {. }$

(Short Answer)

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Question 68

Find the absolute minimum value of the function. $g ( x ) = \frac { e ^ { x } } { x ^ { 4 } } , x > 0$

(Short Answer)

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Question 69

Use the properties of logarithms to expand the quantity. $11 \ln \sqrt { a \left( b ^ { 2 } + c ^ { 2 } \right) }$

(Multiple Choice)

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Question 70

Find the integral. $\int \sqrt { 1 + 8 e ^ { x } } e ^ { x } d x$

(Short Answer)

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Question 71

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln x , y = 1 , y = 5 , x = 0 ;$ about the $y -$ axis

(Multiple Choice)

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Question 72

Simplify the expression. $e ^ { 3 \ln 2 }$

(Multiple Choice)

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Question 73

Evaluate the given integral. $\int _ { 0 } ^ { 1 / 7 } \frac { 1 } { 1 + 49 x ^ { 2 } } d x$

(Multiple Choice)

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Question 74

$\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$ $\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$

(Short Answer)

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Question 75

Find the solution of the equation correct to four decimal places. $e ^ { 1 + 2 x } = 190$

(Short Answer)

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Question 76

Differentiate the function. $y = \ln \left( x ^ { 5 } \sin ^ { 2 } x \right)$

(Short Answer)

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Question 77

Find $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right)$

(Multiple Choice)

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Question 78

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln x , y = 1 , y = 5 , x = 0 ;$ about the $y -$ axis

$\text { Find } y ^ {t } \text { if } \ln x y = 2 y \sin x \text {. }$

Find the points of intersection of the graphs of the functions.Express your answers accurate to five decimal places. $f ( x ) = 0.2 x ^ { 2 } - 1.8 x - 3.8 ; g ( x ) = - 0.1 x ^ { 2 } + 0.8 x + 6.7$

Use the properties of logarithms to expand the quantity. $\log _ { 2 } \left( \frac { x ^ { 10 } y } { z ^ { 5 } } \right)$

Find $\left( f ^ { - 1 } \right) ^ { t } ( a )$ $f ( x ) = 7 + x ^ { 2 } + \tan \left( \frac { \pi x } { 2 } \right) , - 1 < x < 1 , a = 7$

Differentiate the function. $y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)$

Use logarithmic differentiation to find the derivative of the function. $y = x ^ { 6 x }$

$\text { Find } y ^ {t } \text { if } \ln x y = 2 y \sin x \text {. }$

Find the absolute minimum value of the function. $g ( x ) = \frac { e ^ { x } } { x ^ { 4 } } , x > 0$

Use the properties of logarithms to expand the quantity. $11 \ln \sqrt { a \left( b ^ { 2 } + c ^ { 2 } \right) }$

Find the integral. $\int \sqrt { 1 + 8 e ^ { x } } e ^ { x } d x$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln x , y = 1 , y = 5 , x = 0 ;$ about the $y -$ axis

Simplify the expression. $e ^ { 3 \ln 2 }$

Evaluate the given integral. $\int _ { 0 } ^ { 1 / 7 } \frac { 1 } { 1 + 49 x ^ { 2 } } d x$

$\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$ $\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$

Find the solution of the equation correct to four decimal places. $e ^ { 1 + 2 x } = 190$

Differentiate the function. $y = \ln \left( x ^ { 5 } \sin ^ { 2 } x \right)$

Find $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right)$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

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

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Exam 6: Applications of Integration

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=ln⁡x,y=1,y=5,x=0;y = \ln x , y = 1 , y = 5 , x = 0 ;y=lnx,y=1,y=5,x=0; about the y−y -y− axis

Find yt if ln⁡xy=2ysin⁡x. \text { Find } y ^ {t } \text { if } \ln x y = 2 y \sin x \text {. } Find yt if lnxy=2ysinx.

Find the points of intersection of the graphs of the functions.Express your answers accurate to five decimal places. f(x)=0.2x2−1.8x−3.8;g(x)=−0.1x2+0.8x+6.7f ( x ) = 0.2 x ^ { 2 } - 1.8 x - 3.8 ; g ( x ) = - 0.1 x ^ { 2 } + 0.8 x + 6.7f(x)=0.2x2−1.8x−3.8;g(x)=−0.1x2+0.8x+6.7

Use the properties of logarithms to expand the quantity. log⁡2(x10yz5)\log _ { 2 } \left( \frac { x ^ { 10 } y } { z ^ { 5 } } \right)log2​(z5x10y​)

Find (f−1)t(a)\left( f ^ { - 1 } \right) ^ { t } ( a )(f−1)t(a) f(x)=7+x2+tan⁡(πx2),−1<x<1,a=7f ( x ) = 7 + x ^ { 2 } + \tan \left( \frac { \pi x } { 2 } \right) , - 1 < x < 1 , a = 7f(x)=7+x2+tan(2πx​),−1<x<1,a=7

Differentiate the function. y=ln⁡(x3sin⁡2x)y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)y=ln(x3sin2x)

Use logarithmic differentiation to find the derivative of the function. y=x6xy = x ^ { 6 x }y=x6x

Find yt if ln⁡xy=2ysin⁡x. \text { Find } y ^ {t } \text { if } \ln x y = 2 y \sin x \text {. } Find yt if lnxy=2ysinx.

Find the absolute minimum value of the function. g(x)=exx4,x>0g ( x ) = \frac { e ^ { x } } { x ^ { 4 } } , x > 0g(x)=x4ex​,x>0

Use the properties of logarithms to expand the quantity. 11ln⁡a(b2+c2)11 \ln \sqrt { a \left( b ^ { 2 } + c ^ { 2 } \right) }11lna(b2+c2)​

Find the integral. ∫1+8exexdx\int \sqrt { 1 + 8 e ^ { x } } e ^ { x } d x∫1+8ex​exdx

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=ln⁡x,y=1,y=5,x=0;y = \ln x , y = 1 , y = 5 , x = 0 ;y=lnx,y=1,y=5,x=0; about the y−y -y− axis

Simplify the expression. e3ln⁡2e ^ { 3 \ln 2 }e3ln2

Evaluate the given integral. ∫01/711+49x2dx\int _ { 0 } ^ { 1 / 7 } \frac { 1 } { 1 + 49 x ^ { 2 } } d x∫01/7​1+49x21​dx

Find the exponential function f(x)=bax whose graph is given. \text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. } Find the exponential function f(x)=bax whose graph is given.

Find the solution of the equation correct to four decimal places. e1+2x=190e ^ { 1 + 2 x } = 190e1+2x=190

Differentiate the function. y=ln⁡(x5sin⁡2x)y = \ln \left( x ^ { 5 } \sin ^ { 2 } x \right)y=ln(x5sin2x)

Find d4dx4(x3ln⁡x)\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right)dx4d4​(x3lnx)

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

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

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln x , y = 1 , y = 5 , x = 0 ;$ about the $y -$ axis

$\text { Find } y ^ {t } \text { if } \ln x y = 2 y \sin x \text {. }$

Find the points of intersection of the graphs of the functions.Express your answers accurate to five decimal places. $f ( x ) = 0.2 x ^ { 2 } - 1.8 x - 3.8 ; g ( x ) = - 0.1 x ^ { 2 } + 0.8 x + 6.7$

Use the properties of logarithms to expand the quantity. $\log _ { 2 } \left( \frac { x ^ { 10 } y } { z ^ { 5 } } \right)$

Find $\left( f ^ { - 1 } \right) ^ { t } ( a )$ $f ( x ) = 7 + x ^ { 2 } + \tan \left( \frac { \pi x } { 2 } \right) , - 1 < x < 1 , a = 7$

Differentiate the function. $y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)$

Use logarithmic differentiation to find the derivative of the function. $y = x ^ { 6 x }$

$\text { Find } y ^ {t } \text { if } \ln x y = 2 y \sin x \text {. }$

Find the absolute minimum value of the function. $g ( x ) = \frac { e ^ { x } } { x ^ { 4 } } , x > 0$

Use the properties of logarithms to expand the quantity. $11 \ln \sqrt { a \left( b ^ { 2 } + c ^ { 2 } \right) }$

Find the integral. $\int \sqrt { 1 + 8 e ^ { x } } e ^ { x } d x$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln x , y = 1 , y = 5 , x = 0 ;$ about the $y -$ axis

Simplify the expression. $e ^ { 3 \ln 2 }$

Evaluate the given integral. $\int _ { 0 } ^ { 1 / 7 } \frac { 1 } { 1 + 49 x ^ { 2 } } d x$

$\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$ $\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$

Find the solution of the equation correct to four decimal places. $e ^ { 1 + 2 x } = 190$

Differentiate the function. $y = \ln \left( x ^ { 5 } \sin ^ { 2 } x \right)$

Find $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right)$