Question 1

Determine if the function F is the general antiderivative of the function f. F(t) = 5t2 + 3et + C;   f(t) = 10t + 3e

Accepted Answer

A) True 
 B)False  True

Question 2

Refer to the information in the graph below. Set up a definite integral or sum of definite integrals that gives the area of the shaded portion.

Accepted Answer

A)  $\int _ { - 2 } ^ { - 3 x + 4 } \left( 4 - x ^ { 2 } \right) d x$ 
B)  $\int _ { - 2 } ^ { 0 } \left( 4 - x ^ { 2 } \right) d x$ + $\int _ { 0 } ^ { 1 } [ 4 - ( - 3 x + 4 ) ] d x$ 
C)  $\int _ { - 2 } ^ { 1 } \left[ 4 + ( - 3 x + 4 ) - x ^ { 2 } \right] d x$ 
D)  $\int _ { - 2 } ^ { 0 } \left( 4 - x ^ { 2 } \right) d x$ + $\int _ { 0 } ^ { 1 } \left( - 3 x + 4 - x ^ { 2 } \right) d x$ 
E)  none of these 
A)  $\int _ { - 2 } ^ { - 3 x + 4 } \left( 4 - x ^ { 2 } \right) d x$ 
B)  $\int _ { - 2 } ^ { 0 } \left( 4 - x ^ { 2 } \right) d x$ + $\int _ { 0 } ^ { 1 } [ 4 - ( - 3 x + 4 ) ] d x$ 
C)  $\int _ { - 2 } ^ { 1 } \left[ 4 + ( - 3 x + 4 ) - x ^ { 2 } \right] d x$ 
D)  $\int _ { - 2 } ^ { 0 } \left( 4 - x ^ { 2 } \right) d x$ + $\int _ { 0 } ^ { 1 } \left( - 3 x + 4 - x ^ { 2 } \right) d x$ 
E)  none of these D

Question 3

Find the area of the region bounded by y = x and y = $x ^ { 3 }$ .
Enter a reduced fraction $\frac { a } { b }$ .

Accepted Answer

$\frac { 1 } { 2 }$

Question 4

Find all antiderivatives of the function. 
-f(y) = $y ^ { 4 }$ Enter your answer as a polynomial in y in standard form.

Accepted Answer

The answer of Find all antiderivatives of the function. 
-f(y)...

Question 5

$\int _ { 0 } ^ { 1 } \left( e ^ { 3 x - 1 } \right) d x$ Enter your answer as a( $\mathrm { e } ^ { \mathrm { b} }$ ± $\mathrm { e } ^ { \mathrm { C } }$ ). Any fractions in reduced form $\frac { a } { b }$ .

Accepted Answer

The answer of \(\int _ { 0 } ^ {...

Question 6

Given f(x) = $x ^ { 2 }$ + x +1 on the interval 0 ≤ x ≤ 4 and with n = 5, compute the Riemann sum (a) using the left endpoints; (b) using the right endpoints; and (c) using the midpoints of the subintervals. Enter your answer as just a, b, c all integers separated by commas.
Enter the numbers in the order that answers (a), (b), (c) but do not label. Round to the nearest whole number.

Accepted Answer

The answer of Given f(x) = \(x ^ { 2...

Question 7

Find the volume of the solid of revolution generated by revolving the region formed by the graphs of $y = x ^ { 2 }$ $y = 2$ and $x = 0$ about the x-axis.
Enter a reduced quotient $\frac { a b \sqrt { c } } { d }$ .

Accepted Answer

The answer of Find the volume of the solid of...

Question 8

Find the area of the region between y = 3x - 1, the y-axis, and the lines y = 2 and $y = 5$ .

Accepted Answer

A)  $\frac { 26 } { 3 }$ 
B)  $\frac { 17 } { 2 }$ 
C)  $\frac { 19 } { 2 }$ 
D)  $\frac { 9 } { 2 }$ 
E)  none of these 
A)  $\frac { 26 } { 3 }$ 
B)  $\frac { 17 } { 2 }$ 
C)  $\frac { 19 } { 2 }$ 
D)  $\frac { 9 } { 2 }$ 
E)  none of these

Question 9

Find: $\int \left( 2 x ^ { 4 } + 3 x - 4 \right)$ dx
Enter a polynomial in x in standard form with any fractional coefficients or powers in reduced form $\frac { a } { b }$ .

Accepted Answer

The answer of Find: \(\int \left( 2 x ^ {...

Question 10

Given the graph of the function y = $\frac { 1 } { x ^ { 2 } }$ , set up the definite integral that gives the area of the shaded region.

Accepted Answer

A)  $\int _ { 0 } ^ { 2 } \frac { 1 } { x ^ { 2 } }$ dx 
B)  $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } }$ dx 
C)  $\int _ { 2 } ^ { 3 } \frac { 1 } { x ^ { 2 } }$ dx 
D)  $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } }$ dx 
A)  $\int _ { 0 } ^ { 2 } \frac { 1 } { x ^ { 2 } }$ dx 
B)  $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } }$ dx 
C)  $\int _ { 2 } ^ { 3 } \frac { 1 } { x ^ { 2 } }$ dx 
D)  $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } }$ dx

Question 11

Find the area of the region bounded by the curve y = $\left( \frac { 1 } { 2 } x + 3 \right) ^ { - 1 }$ , the y-axis, and the line y = 1.

Accepted Answer

A)  2 $\left( \ln \frac { 7 } { 2 } - \ln 3 \right)$ 
B)  ln 3 
C)  4 -2 ln 3 
D)  ln $\frac { 7 } { 2 }$ - ln 3 
E)  none of these 
A)  2 $\left( \ln \frac { 7 } { 2 } - \ln 3 \right)$ 
B)  ln 3 
C)  4 -2 ln 3 
D)  ln $\frac { 7 } { 2 }$ - ln 3 
E)  none of these

Question 12

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the right endpoints.
Enter just an integer.
f(x) = 2x + 1; 1 ≤ x ≤ 5, n = 4

Accepted Answer

The answer of Use a Riemann sum to approximate the...

Question 13

This is a sketch of the region between the two curves y = $\sqrt { x }$ and y = 2 $\sqrt { x - 1 }$ and the x-axis. Does the following represent the area of the region? $$\int _ { 0 } ^ { 1 } \sqrt { x } d x + \int _ { 1 } ^ { 4 / 3 } \sqrt { x } - 2 \sqrt { x - 1 } d x$$

Accepted Answer

A) True 
 B)False

Question 14

$\int _ { 1 } ^ { 2 } \left( \frac { 1 } { x ^ { 2 } } - 3 \right) d x$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

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

Question 15

Find: $\int \frac { d x } { e ^ { 3 x } }$ Enter your answer in standard form (use a $\mathrm { e } ^ { \mathrm { b} }$ ).

Accepted Answer

The answer of Find: \(\int \frac { d x }...

Question 16

Given the graph of the function y = $x ^ { 2 }$ + 8, set up the definite integral that gives the area of the shaded region.

Accepted Answer

A)  $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right)$ dx 
B)  $\int _ { - 3 } ^ { 2 } \left( x ^ { 2 } + 8 \right)$ dx 
C)  $\int _ { 0 } ^ { 2 } 2 \left( x ^ { 2 } + 8 \right)$ dx 
D)  $\int _ { 2 } ^ { - 3 } \left( x ^ { 2 } + 8 \right)$ dx 
A)  $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right)$ dx 
B)  $\int _ { - 3 } ^ { 2 } \left( x ^ { 2 } + 8 \right)$ dx 
C)  $\int _ { 0 } ^ { 2 } 2 \left( x ^ { 2 } + 8 \right)$ dx 
D)  $\int _ { 2 } ^ { - 3 } \left( x ^ { 2 } + 8 \right)$ dx

Question 17

For the Riemann sum, [5 $\sqrt { 1.4 }$ + 5 $\sqrt { 1.8 }$ + 5 $\sqrt { 2.2 }$ + 5 $\sqrt { 2.6 }$ + 5 $\sqrt { 3 }$ ](0.4); a = 1, determine n, b, and f(x).
Enter your answer as just n, b, f(x) (2 integers in that order separated by commas and followed by a power function in x).

Accepted Answer

The answer of For the Riemann sum, [5 \(\sqrt {...

Question 18

Determine the area under the curve y = 4x + 4 from x = 2 to x = 3.
Enter an integer.

Accepted Answer

The answer of Determine the area under the curve y...

Question 19

This is a sketch of the region between the two curves $y = x ^ { 3 } - 3 x + 1$ and $y = x ^ { 2 } - x + 1$ . Does the following represent the area of the region? $$\int _ { - 1 } ^ { 0 } \left( x ^ { 3 } - x ^ { 2 } - 2 x \right) d x + \int _ { 0 } ^ { 2 } \left( - x ^ { 3 } + x ^ { 2 } + 2 x \right) d x$$

Accepted Answer

A) True 
 B)False

Question 20

Find the value of k that makes the antidifferentiation formula true. 
-$\int ( 7 - x ) ^ { - 1 } d x$ = k ln|7 - x| + C

Accepted Answer

A)  -1 
B)  $\frac { 1 } { 7 }$ 
C)  - $\frac { 1 } { 7 }$ 
D)  1 
A)  -1 
B)  $\frac { 1 } { 7 }$ 
C)  - $\frac { 1 } { 7 }$ 
D)  1

Determine if the function F is the general antiderivative of the function f. F(t) = 5t2 + 3et + C; f(t) = 10t + 3e

Refer to the information in the graph below. Set up a definite integral or sum of definite integrals that gives the area of the shaded portion.

Find the area of the region bounded by y = x and y = $x ^ { 3 }$ . Enter a reduced fraction $\frac { a } { b }$ .

Find all antiderivatives of the function. -f(y) = $y ^ { 4 }$ Enter your answer as a polynomial in y in standard form.

$\int _ { 0 } ^ { 1 } \left( e ^ { 3 x - 1 } \right) d x$ Enter your answer as a( $\mathrm { e } ^ { \mathrm { b} }$ ± $\mathrm { e } ^ { \mathrm { C } }$ ). Any fractions in reduced form $\frac { a } { b }$ .

Find the area of the region between y = 3x - 1, the y-axis, and the lines y = 2 and $y = 5$ .

Find: $\int \left( 2 x ^ { 4 } + 3 x - 4 \right)$ dx Enter a polynomial in x in standard form with any fractional coefficients or powers in reduced form $\frac { a } { b }$ .

Given the graph of the function y = $\frac { 1 } { x ^ { 2 } }$ , set up the definite integral that gives the area of the shaded region. $Given the graph of the function y = \frac { 1 } { x ^ { 2 } } , set up the definite integral that gives the area of the shaded region.$

Find the area of the region bounded by the curve y = $\left( \frac { 1 } { 2 } x + 3 \right) ^ { - 1 }$ , the y-axis, and the line y = 1.

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the right endpoints. Enter just an integer. f(x) = 2x + 1; 1 ≤ x ≤ 5, n = 4

$\int _ { 1 } ^ { 2 } \left( \frac { 1 } { x ^ { 2 } } - 3 \right) d x$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Find: $\int \frac { d x } { e ^ { 3 x } }$ Enter your answer in standard form (use a $\mathrm { e } ^ { \mathrm { b} }$ ).

Given the graph of the function y = $x ^ { 2 }$ + 8, set up the definite integral that gives the area of the shaded region. $Given the graph of the function y = x ^ { 2 } + 8, set up the definite integral that gives the area of the shaded region.$

For the Riemann sum, [5 $\sqrt { 1.4 }$ + 5 $\sqrt { 1.8 }$ + 5 $\sqrt { 2.2 }$ + 5 $\sqrt { 2.6 }$ + 5 $\sqrt { 3 }$ ](0.4); a = 1, determine n, b, and f(x). Enter your answer as just n, b, f(x) (2 integers in that order separated by commas and followed by a power function in x).

Determine the area under the curve y = 4x + 4 from x = 2 to x = 3. Enter an integer.

Find the value of k that makes the antidifferentiation formula true. - $\int ( 7 - x ) ^ { - 1 } d x$ = k ln|7 - x| + C

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 6: The Definite Integral

Determine if the function F is the general antiderivative of the function f. F(t) = 5t2 + 3et + C; f(t) = 10t + 3e

Refer to the information in the graph below. Set up a definite integral or sum of definite integrals that gives the area of the shaded portion.

Find the area of the region bounded by y = x and y = x3x ^ { 3 }x3 . Enter a reduced fraction ab\frac { a } { b }ba​ .

Find all antiderivatives of the function. -f(y) = y4y ^ { 4 }y4 Enter your answer as a polynomial in y in standard form.

∫01(e3x−1)dx\int _ { 0 } ^ { 1 } \left( e ^ { 3 x - 1 } \right) d x∫01​(e3x−1)dx Enter your answer as a( eb\mathrm { e } ^ { \mathrm { b} }eb ± eC\mathrm { e } ^ { \mathrm { C } }eC ). Any fractions in reduced form ab\frac { a } { b }ba​ .

Find the volume of the solid of revolution generated by revolving the region formed by the graphs of y=x2y = x ^ { 2 }y=x2 y=2y = 2y=2 and x=0x = 0x=0 about the x-axis. Enter a reduced quotient abcd\frac { a b \sqrt { c } } { d }dabc​​ .

Find the area of the region between y = 3x - 1, the y-axis, and the lines y = 2 and y=5y = 5y=5 .

Find: ∫(2x4+3x−4)\int \left( 2 x ^ { 4 } + 3 x - 4 \right)∫(2x4+3x−4) dx Enter a polynomial in x in standard form with any fractional coefficients or powers in reduced form ab\frac { a } { b }ba​ .

Given the graph of the function y = 1x2\frac { 1 } { x ^ { 2 } }x21​ , set up the definite integral that gives the area of the shaded region.

Find the area of the region bounded by the curve y = (12x+3)−1\left( \frac { 1 } { 2 } x + 3 \right) ^ { - 1 }(21​x+3)−1 , the y-axis, and the line y = 1.

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the right endpoints. Enter just an integer. f(x) = 2x + 1; 1 ≤ x ≤ 5, n = 4

∫12(1x2−3)dx\int _ { 1 } ^ { 2 } \left( \frac { 1 } { x ^ { 2 } } - 3 \right) d x∫12​(x21​−3)dx Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

Find: ∫dxe3x\int \frac { d x } { e ^ { 3 x } }∫e3xdx​ Enter your answer in standard form (use a eb\mathrm { e } ^ { \mathrm { b} }eb ).

Given the graph of the function y = x2x ^ { 2 }x2 + 8, set up the definite integral that gives the area of the shaded region.

Determine the area under the curve y = 4x + 4 from x = 2 to x = 3. Enter an integer.

Find the value of k that makes the antidifferentiation formula true. - ∫(7−x)−1dx\int ( 7 - x ) ^ { - 1 } d x∫(7−x)−1dx = k ln|7 - x| + C

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Find the area of the region bounded by y = x and y = $x ^ { 3 }$ . Enter a reduced fraction $\frac { a } { b }$ .

Find all antiderivatives of the function. -f(y) = $y ^ { 4 }$ Enter your answer as a polynomial in y in standard form.

$\int _ { 0 } ^ { 1 } \left( e ^ { 3 x - 1 } \right) d x$ Enter your answer as a( $\mathrm { e } ^ { \mathrm { b} }$ ± $\mathrm { e } ^ { \mathrm { C } }$ ). Any fractions in reduced form $\frac { a } { b }$ .

Find the area of the region between y = 3x - 1, the y-axis, and the lines y = 2 and $y = 5$ .

Find: $\int \left( 2 x ^ { 4 } + 3 x - 4 \right)$ dx Enter a polynomial in x in standard form with any fractional coefficients or powers in reduced form $\frac { a } { b }$ .

Given the graph of the function y = $\frac { 1 } { x ^ { 2 } }$ , set up the definite integral that gives the area of the shaded region. $Given the graph of the function y = \frac { 1 } { x ^ { 2 } } , set up the definite integral that gives the area of the shaded region.$

Find the area of the region bounded by the curve y = $\left( \frac { 1 } { 2 } x + 3 \right) ^ { - 1 }$ , the y-axis, and the line y = 1.

$\int _ { 1 } ^ { 2 } \left( \frac { 1 } { x ^ { 2 } } - 3 \right) d x$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Find: $\int \frac { d x } { e ^ { 3 x } }$ Enter your answer in standard form (use a $\mathrm { e } ^ { \mathrm { b} }$ ).

Given the graph of the function y = $x ^ { 2 }$ + 8, set up the definite integral that gives the area of the shaded region. $Given the graph of the function y = x ^ { 2 } + 8, set up the definite integral that gives the area of the shaded region.$

Find the value of k that makes the antidifferentiation formula true. - $\int ( 7 - x ) ^ { - 1 } d x$ = k ln|7 - x| + C