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AP Calculus AB Unit 8B: Volumes of Revolution and Cross-Sections — practice questions

Explore 66 free Unit 8B: Volumes of Revolution and Cross-Sections questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Unit 8B: Volumes of Revolution and Cross-Sections questions and exercises with answers

Showing questions 1–30 of 66.

Which expression correctly represents the vector triple product $\mathbf{A}\times(\mathbf{B}\times\mathbf{C})$?Read the answer and worked solution →If $\mathbf{A}\cdot\mathbf{B}=0$, then the vectors $\mathbf{A}$ and $\mathbf{B}$ areRead the answer and worked solution →Which of the following equals $\mathbf{A}\times\mathbf{B}$ for $\mathbf{A}=\langle 1,2,0\rangle$ and $\mathbf{B}=\langle 3,-1,4\rangle$?Read the answer and worked solution →If $\mathbf{A}=\langle 1,2,3\rangle$, then $3\mathbf{A}$ equalsRead the answer and worked solution →If $\mathbf{A}=\langle 2,-1,4\rangle$ and $\mathbf{B}=\langle 3,5,-2\rangle$, then $\mathbf{A}+\mathbf{B}$ isRead the answer and worked solution →For the matrix equation $AX=B$, if $|A|\neq 0$, the unique solution isRead the answer and worked solution →If $A$ is of order $2\times 3$ and $B$ is of order $3\times 4$, then the product $AB$ is defined and has orderRead the answer and worked solution →Which of the following is a diagonal matrix?Read the answer and worked solution →The direction cosines of a line are $l$, $m$, $n$. They satisfy:Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{2}{x}\right)^x$.Read the answer and worked solution →A line passes through the points $(1, 2, 3)$ and $(4, 6, 9)$. The direction ratios of the line are:Read the answer and worked solution →What is the principal argument of the complex number $z = 1 + i$?Read the answer and worked solution →What is 35² without direct multiplication?Read the answer and worked solution →How many digits will be in the square root of a perfect square with EXACTLY 8 digits?Read the answer and worked solution →How many non-square natural numbers lie between 9² and 10² (exclusive of both)?Read the answer and worked solution →Compute 99² (without long multiplication, using (a + b)² or (a − b)²):Read the answer and worked solution →A square garden has area 2025 m². What is its side?Read the answer and worked solution →Which of these cannot possibly be the LAST digit (units digit) of a perfect square?Read the answer and worked solution →By how much does (n+1)² exceed n²?Read the answer and worked solution →Compute 105² without direct multiplication:Read the answer and worked solution →What is the units digit of (1234)²?Read the answer and worked solution →Using the identity (2m, m² − 1, m² + 1) for Pythagorean triplets, the triplet generated by m = 4 is:Read the answer and worked solution →Solve for z: 3(5z − 7) − 2(9z − 11) = 4(8z − 13) − 17Read the answer and worked solution →The square root of 0.81 is:Read the answer and worked solution →Square root of 144 is:Read the answer and worked solution →A Pythagorean triplet has one member equal to 12, and the OTHER TWO members differ by 2. The triplet is:Read the answer and worked solution →What is the smallest natural number by which 180 must be MULTIPLIED so that the product becomes a perfect square?Read the answer and worked solution →What is the smallest natural number by which 1620 must be DIVIDED so that the quotient is a perfect square?Read the answer and worked solution →Compute the square root of 12.96 (a decimal).Read the answer and worked solution →Two square fields have areas in the ratio 16 : 9. The ratio of their PERIMETERS is:Read the answer and worked solution →

Worked practice questions

Which expression correctly represents the vector triple product $\mathbf{A}\times(\mathbf{B}\times\mathbf{C})$?

  1. $(\mathbf{A}\cdot\mathbf{B})\mathbf{C}-(\mathbf{A}\cdot\mathbf{C})\mathbf{B}$
  2. $(\mathbf{B}\cdot\mathbf{C})\mathbf{A}-(\mathbf{A}\cdot\mathbf{C})\mathbf{B}$
  3. $(\mathbf{A}\cdot\mathbf{C})\mathbf{B}-(\mathbf{A}\cdot\mathbf{B})\mathbf{C}$
  4. $(\mathbf{A}\times\mathbf{B})\cdot\mathbf{C}$
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Correct answer: C. $(\mathbf{A}\cdot\mathbf{C})\mathbf{B}-(\mathbf{A}\cdot\mathbf{B})\mathbf{C}$

The standard identity is $\mathbf{A}\times(\mathbf{B}\times\mathbf{C})=(\mathbf{A}\cdot\mathbf{C})\mathbf{B}-(\mathbf{A}\cdot\mathbf{B})\mathbf{C}$. This is often remembered as the BAC-CAB rule.

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If $\mathbf{A}\cdot\mathbf{B}=0$, then the vectors $\mathbf{A}$ and $\mathbf{B}$ are

  1. parallel
  2. equal in magnitude
  3. perpendicular
  4. collinear
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Correct answer: C. perpendicular

A zero dot product indicates orthogonality, so the vectors are perpendicular. Parallel or collinear vectors generally have a nonzero dot product unless one vector is zero.

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Which of the following equals $\mathbf{A}\times\mathbf{B}$ for $\mathbf{A}=\langle 1,2,0\rangle$ and $\mathbf{B}=\langle 3,-1,4\rangle$?

  1. $\langle 8,-4,-7\rangle$
  2. $\langle -8,4,7\rangle$
  3. $\langle 8,4,-7\rangle$
  4. $\langle -8,-4,7\rangle$
Show answer and solution

Correct answer: A. $\langle 8,-4,-7\rangle$

Using $\mathbf{A}\times\mathbf{B}=\langle A_2B_3-A_3B_2,\ A_3B_1-A_1B_3,\ A_1B_2-A_2B_1\rangle$, we get $\langle 2\cdot 4-0\cdot(-1),\ 0\cdot 3-1\cdot 4,\ 1\cdot(-1)-2\cdot 3\rangle=\langle 8,-4,-7\rangle$. A common mistake is reversing the order, which would change the sign.

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 3A: Chain Rule and Composite FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 2: Differentiation and Fundamental Derivative RulesUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite Limits