allery Start learning →

AP Calculus AB Unit 8A: Areas Between Curves — practice questions — page 2

Explore 66 free Unit 8A: Areas Between Curves questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

Practise Unit 8A: Areas Between Curves with Quill →

Unit 8A: Areas Between Curves questions and exercises with answers

Showing questions 31–60 of 66.

The direction cosines of a line whose direction ratios are $2,-1,2$ are:Read the answer and worked solution →The cosine of the angle between two lines with direction ratios $(1,2,2)$ and $(2,2,1)$ is:Read the answer and worked solution →The cartesian equation of a line through $(x_1,y_1,z_1)$ with direction ratios $a,b,c$ is:Read the answer and worked solution →A line makes equal angles with the three coordinate axes. Its direction cosines are:Read the answer and worked solution →A line with direction ratios $(k,2,3)$ is perpendicular to a line with direction ratios $(1,-2,1)$. Then $k$ equals:Read the answer and worked solution →The distance of the point $(2,3,4)$ from the origin is:Read the answer and worked solution →If a line makes angles $90^\circ, 60^\circ, 30^\circ$ with the x, y, z axes, the sum of squares of its direction cosines is:Read the answer and worked solution →A hyperbola has foci at $(\pm 5, 0)$ and the length of the transverse axis is $6$. Find its equation.Read the answer and worked solution →The distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in three-dimensional space is:Read the answer and worked solution →A square matrix $A$ is orthogonal whenRead the answer and worked solution →The standard equation of a hyperbola with centre at the origin and foci on the $x$-axis is:Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to 0} \dfrac{\sin 3x}{x}$.Read the answer and worked solution →A fair coin is tossed twice. How many outcomes are there in the sample space?Read the answer and worked solution →Find the equation of the parabola with vertex at the origin and focus at $(0, -3)$.Read the answer and worked solution →What is the eccentricity of the hyperbola $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$?Read the answer and worked solution →Consider $f(x) = |x|$ at $x = 0$. Which statement is correct?Read the answer and worked solution →Using De Moivre's theorem, evaluate $\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)^6$.Read the answer and worked solution →Find the length of the latus rectum of the parabola $y^2 = 8x$.Read the answer and worked solution →A square matrix $A$ is symmetric ifRead the answer and worked solution →For any hyperbola, the eccentricity $e$ satisfies:Read the answer and worked solution →A parabola has its vertex at the origin, its axis along the positive $x$-axis, and passes through the point $(4, 6)$. Find its equation.Read the answer and worked solution →If $z = \cos\theta + i\sin\theta$, find an expression for $z^n + \dfrac{1}{z^n}$.Read the answer and worked solution →Find the eccentricity of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$.Read the answer and worked solution →The foci of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ lie at:Read the answer and worked solution →An ellipse has foci at $(\pm 4, 0)$ and passes through the point $(5, 0)$. Find its standard equation.Read the answer and worked solution →For the parabola $y^2 = 16x$, find the equation of the directrix.Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$.Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos 2x}{x^2}$.Read the answer and worked solution →The locus of the point $z$ in the Argand plane satisfying $|z - 2| = |z + 2|$ is:Read the answer and worked solution →Find the modulus of the complex number $z = 3 + 4i$.Read the answer and worked solution →

Worked practice questions

The direction cosines of a line whose direction ratios are $2,-1,2$ are:

$2, -1, 2$
$\tfrac12, -\tfrac11, \tfrac12$
$\tfrac29, -\tfrac19, \tfrac29$
$\tfrac23, -\tfrac13, \tfrac23$

Show answer and solution

Correct answer: D. $\tfrac23, -\tfrac13, \tfrac23$

Divide by √(2²+1²+2²)=3: direction cosines are (2/3, −1/3, 2/3).

Open this question →

The cosine of the angle between two lines with direction ratios $(1,2,2)$ and $(2,2,1)$ is:

$\tfrac{4}{9}$
$\tfrac{2}{3}$
$\tfrac{1}{9}$
$\tfrac{8}{9}$

Show answer and solution

Correct answer: D. $\tfrac{8}{9}$

We have cosθ = (1·2+2·2+2·1)/(√9·√9). So cosθ = 8/9.

Open this question →

The cartesian equation of a line through $(x_1,y_1,z_1)$ with direction ratios $a,b,c$ is:

$\tfrac{x-x_1}{a}+\tfrac{y-y_1}{b}+\tfrac{z-z_1}{c}=1$
$\tfrac{a}{x-x_1}=\tfrac{b}{y-y_1}=\tfrac{c}{z-z_1}$
$\tfrac{x-x_1}{a}=\tfrac{y-y_1}{b}=\tfrac{z-z_1}{c}$
$ax+by+cz=0$

Show answer and solution

Correct answer: C. $\tfrac{x-x_1}{a}=\tfrac{y-y_1}{b}=\tfrac{z-z_1}{c}$

The symmetric (cartesian) form of a line is (x−x₁)/a = (y−y₁)/b = (z−z₁)/c.

Open this question →

How to practise

Choose a Unit 8A: Areas Between Curves question and try to work out your answer.
Open the question to check the correct answer and read its worked solution.
Continue in the app for bite-sized lessons, explanations and review with Quill. You can start without creating an account.

Put it into practice

Continue with Unit 8A: Areas Between Curves practice for AP Calculus AB. Read explanations as you go. No signup form needed to start.

Practise Unit 8A: Areas Between Curves →

More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse Functions Unit 3A: Chain Rule and Composite Functions Unit 1A3: Function Composition and Inverses Unit 1A1: Relations, Functions and Graphs Unit 1A2: Trigonometric Functions & Inverse Trig Unit 2: Differentiation and Fundamental Derivative Rules Unit 6A: Indefinite Integrals & Fundamental Theorem Unit 1B3: Asymptotic Behavior and Infinite Limits