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

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.

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

Showing questions 1–30 of 66.

In 3D, distance between points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):Read the answer and worked solution →Direction cosines (l, m, n) of a line satisfy:Read the answer and worked solution →Equation of line through point (x₀, y₀, z₀) with direction (a, b, c):Read the answer and worked solution →Two lines are parallel if their direction ratios are:Read the answer and worked solution →Normal vector to plane ax + by + cz = d is:Read the answer and worked solution →Distance from point (1, 2, 3) to plane x + y + z = 0:Read the answer and worked solution →Find equation of plane through (1, 0, 0), (0, 2, 0), (0, 0, 3):Read the answer and worked solution →Two planes are perpendicular if:Read the answer and worked solution →Vector form of line: r = a + λ b where:Read the answer and worked solution →Coplanar lines have:Read the answer and worked solution →Equation of sphere with center (h, k, l) and radius r:Read the answer and worked solution →Angle between line and plane: if direction of line is L and normal to plane is N, sin(angle line-plane):Read the answer and worked solution →Projection of vector a on b:Read the answer and worked solution →Skew lines in 3D are:Read the answer and worked solution →Distance between two parallel planes 2x + y + z = 5 and 2x + y + z = 10:Read the answer and worked solution →Find equation of plane perpendicular to (1, 2, 3) passing through (4, 5, 6):Read the answer and worked solution →Angle θ between two planes 2x + 3y + 4z = 0 and x + y + z = 0:Read the answer and worked solution →Three points (1,0,0), (0,1,0), (0,0,1) are vertices of a triangle. Area:Read the answer and worked solution →Plane passing through origin and perpendicular to line (x-1)/2 = (y-2)/3 = (z-3)/-1:Read the answer and worked solution →Foot of perpendicular from point (3, 1, 2) to plane x + y + z = 6:Read the answer and worked solution →Direction ratios of normal to plane 2x - y + 3z + 7 = 0:Read the answer and worked solution →For a line with direction cosines l, m, n, the angles it makes with axes are:Read the answer and worked solution →Coordinates of foot of perpendicular from point (1, 2, 3) to x-axis:Read the answer and worked solution →Equation of straight line passing through two points A(1,2,3) and B(4,5,6):Read the answer and worked solution →3D analog of Pythagoras: for a box with sides a, b, c, the body diagonal is:Read the answer and worked solution →If $l, m, n$ are the direction cosines of a line, then $l^2+m^2+n^2$ equals:Read the answer and worked solution →The direction ratios of the line joining the points $(1,2,3)$ and $(4,5,6)$ are:Read the answer and worked solution →The direction cosines of the x-axis are:Read the answer and worked solution →Two lines are parallel if and only if their direction ratios are:Read the answer and worked solution →The distance between the points $(1,2,3)$ and $(4,5,6)$ is:Read the answer and worked solution →

Worked practice questions

In 3D, distance between points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):

  1. sqrt(x²+y²+z²)
  2. |x₁-x₂|
  3. sqrt((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)
  4. x₁ + x₂
Show answer and solution

Correct answer: C. sqrt((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)

3D distance formula: extension of 2D Pythagoras. d = sqrt(Δx² + Δy² + Δz²).

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Direction cosines (l, m, n) of a line satisfy:

  1. l = m = n
  2. l² + m² + n² = 1 (unit vector property)
  3. Always 0
  4. l + m + n = 1
Show answer and solution

Correct answer: B. l² + m² + n² = 1 (unit vector property)

**Direction cosines:** cosines of angles the line makes with x, y, z axes. (l, m, n) form a unit vector along the line; hence l² + m² + n² = 1.

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Equation of line through point (x₀, y₀, z₀) with direction (a, b, c):

  1. Cannot write
  2. (x - x₀)/a = (y - y₀)/b = (z - z₀)/c (symmetric form)
  3. Only one equation
  4. Same as 2D
Show answer and solution

Correct answer: B. (x - x₀)/a = (y - y₀)/b = (z - z₀)/c (symmetric form)

**Symmetric form:** (x - x₀)/a = (y - y₀)/b = (z - z₀)/c = t (parameter). **Equivalent to parametric:** x = x₀ + at, etc.

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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