Question 1

Evaluate the following integrals:
(a) $\int x ^ { 2 } \ln x d x$ (b) $\int x ( x + 1 ) ^ { 4 } d x$ (c) $\int x ^ { 2 } e ^ { x } d x$ (d) $\int x \cos ( 2 x ) d x$

Accepted Answer

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

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 left endpoint 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

Use the Trapezoidal Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$

Accepted Answer

A)  $\frac { 65 } { 4 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 45 } { 8 }$ 
D)  $\frac { 3 } { 2 }$ 
E)  $\frac { 65 } { 12 }$ 
F) $\frac { 45 } { 4 }$ 
G) $\frac { 7 } { 16 }$ 
H) $\frac { 5 } { 2 }$ 
A)  $\frac { 65 } { 4 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 45 } { 8 }$ 
D)  $\frac { 3 } { 2 }$ 
E)  $\frac { 65 } { 12 }$ 
F) $\frac { 45 } { 4 }$ 
G) $\frac { 7 } { 16 }$ 
H) $\frac { 5 } { 2 }$

Question 4

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

Accepted Answer

A)  $\frac { 3776 } { 3465 }$ 
B)  $\frac { 7552 } { 3465 }$ 
C)  $\frac { 1888 } { 3465 }$ 
D)  $\frac { 7 } { 32 }$ 
E)  $\frac { 5 } { 16 }$ 
F) $\frac { 5 } { 32 }$ 
G) $\frac { 7 } { 16 }$ 
H) $\frac { 5 } { 2 }$ 
A)  $\frac { 3776 } { 3465 }$ 
B)  $\frac { 7552 } { 3465 }$ 
C)  $\frac { 1888 } { 3465 }$ 
D)  $\frac { 7 } { 32 }$ 
E)  $\frac { 5 } { 16 }$ 
F) $\frac { 5 } { 32 }$ 
G) $\frac { 7 } { 16 }$ 
H) $\frac { 5 } { 2 }$

Question 5

Evaluate the following integrals:
(a) $\int \frac { x } { \left( x ^ { 2 } + 1 \right) ^ { 4 } } d x$ (b) $\int _ { e } ^ { e ^ { 2 } } \frac { 1 } { x ( \ln x ) ^ { 2 } } d x$ (c) $\int \frac { e ^ { 1 / x } } { x ^ { 2 } } d x$ (d) $\int _ { 0 } ^ { 1 } \frac { x } { x + 1 } d x$

Accepted Answer

The answer of Evaluate the following integrals:
(a) \(\int \frac {...

Question 6

Find the value of the integral $\int _ { 0 } ^ { 1 } \ln \left( 1 + x ^ { 2 } \right) d x$

Accepted Answer

A)  $\ln 2$ 
B)  $\frac { \pi } { 8 }$ 
C)  $\frac { \pi } { 2 } - 2 + \ln 2$ 
D)  $2 - \ln 2$ 
E)  $\frac { \pi } { 4 } + \ln 2$ 
F) $\pi - 4$

G) $\pi - 2$

H) $\pi - \ln 2$ 
A)  $\ln 2$ 
B)  $\frac { \pi } { 8 }$ 
C)  $\frac { \pi } { 2 } - 2 + \ln 2$ 
D)  $2 - \ln 2$ 
E)  $\frac { \pi } { 4 } + \ln 2$ 
F) $\pi - 4$

G) $\pi - 2$

H) $\pi - \ln 2$

Question 7

The velocity v (in cm/s) of a rolling ball at time t (in seconds) is recorded below. Find an upper and lower estimate for the distance the ball has traveled during the ten-second interval.$\begin{array} { | c | c | c | c | c | c | c | c | c | c | c | c | } 
\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \
\hline v & 1 & 2.9 & 6.6 & 12.1 & 19.4 & 28.5 & 39.4 & 52.1 & 66.6 & 82.9 & 101 \
\hline
\end{array}$

Accepted Answer

Lower estimate @#IMG-DLM& 311.

Question 8

Use the Comparison Properties of Integrals to show that $\frac { 2 } { 9 } \leq \int _ { 0 } ^ { 2 } \frac { 1 } { 1 + x ^ { 3 } } \leq 2$

Accepted Answer

@#IMG-DLM& since @#IMG-DLM& is decreasin

Question 9

What is the value of the estimate using four approximating rectangles and taking sample points to be midpoints?

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 7 } { 4 }$ 
C)  $\frac { 21 } { 8 }$ 
D) 3 
E)  $\frac { 15 } { 4 }$ 
F) $\frac { 29 } { 8 }$ 
 
G) $\frac { 9 } { 2 }$ 
H)5 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 7 } { 4 }$ 
C)  $\frac { 21 } { 8 }$ 
D) 3 
E)  $\frac { 15 } { 4 }$ 
F) $\frac { 29 } { 8 }$ 
 
G) $\frac { 9 } { 2 }$ 
H)5

Question 10

Find the value of the integral $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$

Accepted Answer

A) 1 
B)  $\frac { \pi } { 16 }$ 
C)  $\frac { \pi } { 4 }$ 
D) 2 
E)  $\frac { \pi } { 32 }$ 
F) $\frac { \pi } { 8 }$ 
G) $\frac { \pi } { 64 }$ 
H) $\frac { 1 } { 6 }$ 
A) 1 
B)  $\frac { \pi } { 16 }$ 
C)  $\frac { \pi } { 4 }$ 
D) 2 
E)  $\frac { \pi } { 32 }$ 
F) $\frac { \pi } { 8 }$ 
G) $\frac { \pi } { 64 }$ 
H) $\frac { 1 } { 6 }$

Question 11

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 right endpoint 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 12

Evaluate the improper integral $\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } } d x$

Accepted Answer

A)  $e$ 
B)  $e ^ { 2 } - 1$ 
C) 0 
D)  $e ^ { 2 }$ 
E)  $e ^ { - 1 }$ 
F) $e ^ { - 2 }$ 
G)1
H) $1 - e ^ { - 2 }$ 
A)  $e$ 
B)  $e ^ { 2 } - 1$ 
C) 0 
D)  $e ^ { 2 }$ 
E)  $e ^ { - 1 }$ 
F) $e ^ { - 2 }$ 
G)1
H) $1 - e ^ { - 2 }$

Question 13

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

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 3 } { 2 }$ 
E)  $\frac { 5 } { 4 }$  
F) $\frac { 5 } { 2 }$

G)1

H)0 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 3 } { 2 }$ 
E)  $\frac { 5 } { 4 }$  
F) $\frac { 5 } { 2 }$

G)1

H)0

Question 14

Consider the integral $\int _ { 0 } ^ { 05 } \cos \left( x ^ { 3 } \right) d x$ Approximating it by the Midpoint Rule with n equal subintervals, give an estimate for n which guarantees that the error is bounded by $\frac { 1 } { 10 ^ { 4 } } .$

Accepted Answer

The answer of Consider the integral \(\int _ { 0...

Question 15

A stone is dropped off a bridge into a still pond. The stone has initial velocity of zero and falls increasingly faster under the pull of gravity. When it enters the water it slows down at a constant rate, coming to a terminal velocity then continuing at that rate until it comes to rest on the bottom of the pond.(a) Sketch a rough graph of the stone's velocity as a function of time.(b) Sketch a rough graph of the stone's vertical distance from the point from which it was dropped as a function of time.

Accepted Answer

The answer of A stone is dropped off a bridge...

Question 16

Evaluate the following integrals:
(a) $\int x \cot ^ { - 1 } x d x$ (b) $\int _ { 1 } ^ { e } ( \ln x ) ^ { 2 } d x$ (c) $\int x ^ { 2 } \cos x d x$ (d) $\int \cos ( \ln x ) d x$

Accepted Answer

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

Question 17

At the Dr. T. Bottling Company, the amount of soda dispensed into the cans is normally distributed so the probability that a can will contain a certain amount of soda may be calculated using the area under the curve y = p (x), where p(x) = $\frac { 1 } { \sqrt { 2 \pi } } e ^ { - x ^ { 2 } / 2 }$ Use your graphing calculator to produce a graph of this function.(a) The probability that a can will contain between 12.0 and 12.2 ounces of soda is given by $\int _ { - 0.22 } ^ { 0.22 } \frac { 1 } { \sqrt { 2 \pi } } e ^ { - x ^ { 2 } / 2 } d x$ Use Simpson's Rule with n = 4 to approximate this probability.(b) The probability that a can will contain at least 12 ounces of soda is given by $\int _ { - 0.22 } ^ { \infty } \frac { 1 } { \sqrt { 2 \pi } } e ^ { - x ^ { 2 } / 2 } d x$ What is the approximate value of this integral?

Accepted Answer

(a) About

Question 18

The area of the region under the $f ( x ) = \frac { 1 } { 4 + x ^ { 2 } }$ is defined to be $\int _ { - \infty } ^ { \infty } f ( x ) d x$ Find this area.

Accepted Answer

The answer of The area of the region under the...

Question 19

Evaluate the following integrals:
(a) $\int _ { 0 } ^ { 1 / 2 } \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x$ (b) $\int e ^ { x + e ^ { x } } d x$ (c) $\int \frac { 3 x + 2 } { x - 1 } d x$ (d) $\int \ln \left( 1 + x ^ { 2 } \right) d x$

Accepted Answer

The answer of Evaluate the following integrals:
(a) \(\int _ {...

Question 20

Find the value of the integral $\int _ { 1 } ^ { 2 } \sqrt { u ^ { 2 } - 1 } d u$

Accepted Answer

A)  $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 4 - \sqrt { 3 } )$ 
B)  $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 2 + \sqrt { 3 } )$ 
C)  $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 4 + \sqrt { 3 } )$ 
D)  $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 2 - \sqrt { 3 } )$ 
E)  $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 4 - \sqrt { 3 } )$ 
F) $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 4 + \sqrt { 3 } )$ 
G) $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 2 + \sqrt { 3 } )$ 
H) $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 2 - \sqrt { 3 } )$ 
A)  $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 4 - \sqrt { 3 } )$ 
B)  $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 2 + \sqrt { 3 } )$ 
C)  $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 4 + \sqrt { 3 } )$ 
D)  $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 2 - \sqrt { 3 } )$ 
E)  $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 4 - \sqrt { 3 } )$ 
F) $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 4 + \sqrt { 3 } )$ 
G) $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 2 + \sqrt { 3 } )$ 
H) $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 2 - \sqrt { 3 } )$

Evaluate the following integrals: (a) $\int x ^ { 2 } \ln x d x$ (b) $\int x ( x + 1 ) ^ { 4 } d x$ (c) $\int x ^ { 2 } e ^ { x } d x$ (d) $\int x \cos ( 2 x ) d x$

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 left endpoint of its subinterval.

Use the Trapezoidal Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$

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

Evaluate the following integrals: (a) $\int \frac { x } { \left( x ^ { 2 } + 1 \right) ^ { 4 } } d x$ (b) $\int _ { e } ^ { e ^ { 2 } } \frac { 1 } { x ( \ln x ) ^ { 2 } } d x$ (c) $\int \frac { e ^ { 1 / x } } { x ^ { 2 } } d x$ (d) $\int _ { 0 } ^ { 1 } \frac { x } { x + 1 } d x$

Find the value of the integral $\int _ { 0 } ^ { 1 } \ln \left( 1 + x ^ { 2 } \right) d x$

The velocity v (in cm/s) of a rolling ball at time t (in seconds) is recorded below. Find an upper and lower estimate for the distance the ball has traveled during the ten-second interval. t 0 1 2 3 4 5 6 7 8 9 10 v 1 2.9 6.6 12.1 19.4 28.5 39.4 52.1 66.6 82.9 101

Use the Comparison Properties of Integrals to show that $\frac { 2 } { 9 } \leq \int _ { 0 } ^ { 2 } \frac { 1 } { 1 + x ^ { 3 } } \leq 2$

What is the value of the estimate using four approximating rectangles and taking sample points to be midpoints?

Find the value of the integral $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$

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 right endpoint of its subinterval.

Evaluate the improper integral $\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } } d x$

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

Consider the integral $\int _ { 0 } ^ { 05 } \cos \left( x ^ { 3 } \right) d x$ Approximating it by the Midpoint Rule with n equal subintervals, give an estimate for n which guarantees that the error is bounded by $\frac { 1 } { 10 ^ { 4 } } .$

Evaluate the following integrals: (a) $\int x \cot ^ { - 1 } x d x$ (b) $\int _ { 1 } ^ { e } ( \ln x ) ^ { 2 } d x$ (c) $\int x ^ { 2 } \cos x d x$ (d) $\int \cos ( \ln x ) d x$

The area of the region under the $f ( x ) = \frac { 1 } { 4 + x ^ { 2 } }$ is defined to be $\int _ { - \infty } ^ { \infty } f ( x ) d x$ Find this area.

Evaluate the following integrals: (a) $\int _ { 0 } ^ { 1 / 2 } \frac { x } { \sqrt { 1 - x ^ { 2 } } } d x$ (b) $\int e ^ { x + e ^ { x } } d x$ (c) $\int \frac { 3 x + 2 } { x - 1 } d x$ (d) $\int \ln \left( 1 + x ^ { 2 } \right) d x$

Find the value of the integral $\int _ { 1 } ^ { 2 } \sqrt { u ^ { 2 } - 1 } d u$

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Exam 5: Integrals

Evaluate the following integrals: (a) ∫x2ln⁡xdx\int x ^ { 2 } \ln x d x∫x2lnxdx (b) ∫x(x+1)4dx\int x ( x + 1 ) ^ { 4 } d x∫x(x+1)4dx (c) ∫x2exdx\int x ^ { 2 } e ^ { x } d x∫x2exdx (d) ∫xcos⁡(2x)dx\int x \cos ( 2 x ) d x∫xcos(2x)dx

Use the Trapezoidal Rule with n = 4 to approximate the integral ∫−222xdx\int _ { - 2 } ^ { 2 } 2 ^ { x } d x∫−22​2xdx

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

Find the value of the integral ∫01ln⁡(1+x2)dx\int _ { 0 } ^ { 1 } \ln \left( 1 + x ^ { 2 } \right) d x∫01​ln(1+x2)dx

The velocity v (in cm/s) of a rolling ball at time t (in seconds) is recorded below. Find an upper and lower estimate for the distance the ball has traveled during the ten-second interval. t 0 1 2 3 4 5 6 7 8 9 10 v 1 2.9 6.6 12.1 19.4 28.5 39.4 52.1 66.6 82.9 101

Use the Comparison Properties of Integrals to show that 29≤∫0211+x3≤2\frac { 2 } { 9 } \leq \int _ { 0 } ^ { 2 } \frac { 1 } { 1 + x ^ { 3 } } \leq 292​≤∫02​1+x31​≤2

What is the value of the estimate using four approximating rectangles and taking sample points to be midpoints?

Find the value of the integral ∫0π/4sin⁡2(2x)cos⁡2(2x)dx\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x∫0π/4​sin2(2x)cos2(2x)dx

Evaluate the improper integral ∫−∞∞xe−x2dx\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } } d x∫−∞∞​xe−x2dx

Find the value of the integral ∫−20∣x+1∣dx\int _ { - 2 } ^ { 0 } | x + 1 | d x∫−20​∣x+1∣dx

Consider the integral ∫005cos⁡(x3)dx\int _ { 0 } ^ { 05 } \cos \left( x ^ { 3 } \right) d x∫005​cos(x3)dx Approximating it by the Midpoint Rule with n equal subintervals, give an estimate for n which guarantees that the error is bounded by 1104.\frac { 1 } { 10 ^ { 4 } } .1041​.

Evaluate the following integrals: (a) ∫xcot⁡−1xdx\int x \cot ^ { - 1 } x d x∫xcot−1xdx (b) ∫1e(ln⁡x)2dx\int _ { 1 } ^ { e } ( \ln x ) ^ { 2 } d x∫1e​(lnx)2dx (c) ∫x2cos⁡xdx\int x ^ { 2 } \cos x d x∫x2cosxdx (d) ∫cos⁡(ln⁡x)dx\int \cos ( \ln x ) d x∫cos(lnx)dx

The area of the region under the f(x)=14+x2f ( x ) = \frac { 1 } { 4 + x ^ { 2 } }f(x)=4+x21​ is defined to be ∫−∞∞f(x)dx\int _ { - \infty } ^ { \infty } f ( x ) d x∫−∞∞​f(x)dx Find this area.

Find the value of the integral ∫12u2−1du\int _ { 1 } ^ { 2 } \sqrt { u ^ { 2 } - 1 } d u∫12​u2−1​du

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Evaluate the following integrals: (a) $\int x ^ { 2 } \ln x d x$ (b) $\int x ( x + 1 ) ^ { 4 } d x$ (c) $\int x ^ { 2 } e ^ { x } d x$ (d) $\int x \cos ( 2 x ) d x$

Use the Trapezoidal Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$

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

Find the value of the integral $\int _ { 0 } ^ { 1 } \ln \left( 1 + x ^ { 2 } \right) d x$

Use the Comparison Properties of Integrals to show that $\frac { 2 } { 9 } \leq \int _ { 0 } ^ { 2 } \frac { 1 } { 1 + x ^ { 3 } } \leq 2$

Find the value of the integral $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$

Evaluate the improper integral $\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } } d x$

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

Consider the integral $\int _ { 0 } ^ { 05 } \cos \left( x ^ { 3 } \right) d x$ Approximating it by the Midpoint Rule with n equal subintervals, give an estimate for n which guarantees that the error is bounded by $\frac { 1 } { 10 ^ { 4 } } .$

Evaluate the following integrals: (a) $\int x \cot ^ { - 1 } x d x$ (b) $\int _ { 1 } ^ { e } ( \ln x ) ^ { 2 } d x$ (c) $\int x ^ { 2 } \cos x d x$ (d) $\int \cos ( \ln x ) d x$

The area of the region under the $f ( x ) = \frac { 1 } { 4 + x ^ { 2 } }$ is defined to be $\int _ { - \infty } ^ { \infty } f ( x ) d x$ Find this area.

Find the value of the integral $\int _ { 1 } ^ { 2 } \sqrt { u ^ { 2 } - 1 } d u$