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AP®MathsCollege BoardCalculus BCExam QuestionsUnit 8: Applications of IntegrationAreas & Arc Lengths

Areas & Arc Lengths (College Board AP® Calculus BC): Exam Questions

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Areas & Arc Lengths

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1
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Graph of the quadratic function y = -3x² + 6x + 9, a downward opening parabola, shaded area between x = -1 and x = 3 on a coordinate plane.

The diagram shows the graph of y equals negative 3 x squared plus 6 x plus 9 crossing the x-axis at the points open parentheses negative 1 comma space 0 close parentheses and open parentheses 3 comma space 0 close parentheses. The area of the shaded region is

  • 32

  • 34

  • 38

  • 40

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2
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What is the area of the region in the first quadrant bounded by the graph of y equals e to the power of x over 3 end exponent and the line x equals 3?

  • 3 e minus 3

  • 3 e

  • e over 3 minus 1

  • fraction numerator e minus 1 over denominator 3 end fraction

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3
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Graph of y = cos x from 0 to 5\pi/2 with a shaded area under the curve from 0 to \pi/2, showing peaks and troughs.

If the shaded region has an area of 1, what is the value of integral subscript 0 superscript fraction numerator 5 straight pi over denominator 2 end fraction end superscript cos space x space d x?

  • 5

  • 3

  • 1

  • 0

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4
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Graph of y = \sqrt{x+1} with a shaded area to the left of the curve, between y=1 and y=2, bounded by dashed line, axes labelled x and y.

The area of the region in the first quadrant shown above, bounded by the curve y equals square root of x plus 1 end root , the y-axis and the line y equals 2, is

  • 10 over 3

  • 4 over 3

  • 2 over 3

  • negative 2 over 3

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5
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Graph of y = x^4 - 10x^3 + 32x^2 - 32x, showing a shaded area under a peak above the x-axis.

The area of the shaded region between the graph of y equals x to the power of 4 minus 10 x cubed plus 32 x squared minus 32 x and the x-axis is

  • 16

  • 12.267

  • 8.533

  • 3.733

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1
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If 0 less or equal than k less than pi and the area under the curve y equals sin x from x equals 0 to x equals k is 1.2, then k equals

  • 0.201

  • 1.772

  • 2.940

  • 4.511

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2
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The function f is defined by f open parentheses x close parentheses equals open curly brackets table row cell 5 minus x space space space for space x less or equal than 2 end cell row cell 3 space space space space space space space space space for space x greater than 2 end cell end table close. What is the value of integral subscript negative 1 end subscript superscript 6 f open parentheses x close parentheses space d x?

  • 21

  • 23.5

  • 25.5

  • 27

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3
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The area of the region enclosed by the graph of y equals x squared minus 2 x plus 1 and the graph of y equals x plus 1 is

  • 7 over 6

  • 3

  • 9 over 2

  • 21 over 2

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4
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Graph of y = f(x) with shaded area under curve from x=0 to x=5. Curve rises, falls, and rises again with axes labelled x and y.

Which one option out of the following does not represent the total shaded area shown above?

  • integral subscript 0 superscript 3 f open parentheses x close parentheses space d x plus open vertical bar integral subscript 3 superscript 5 f open parentheses x close parentheses space d x close vertical bar

  • integral subscript 0 superscript 3 f open parentheses x close parentheses space d x plus integral subscript 3 superscript 5 f open parentheses x close parentheses space d x

  • integral subscript 0 superscript 3 f open parentheses x close parentheses space d x minus integral subscript 3 superscript 5 f open parentheses x close parentheses space d x

  • integral subscript 0 superscript 5 open vertical bar f open parentheses x close parentheses close vertical bar space d x

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5
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Graph of y = 0.5 ln((x-5)/4), with shaded area to the left of the curve.

The area of the shaded region between y equals 1 half ln open parentheses fraction numerator x minus 5 over denominator 4 end fraction close parentheses and the y-axis, shown above, is

  • integral subscript 0 superscript 21 space 1 half ln open parentheses fraction numerator x minus 5 over denominator 4 end fraction close parentheses space italic d x

  • integral subscript 9 superscript 21 space 1 half ln open parentheses fraction numerator x minus 5 over denominator 4 end fraction close parentheses space italic d x

  • integral subscript 9 superscript 21 open parentheses 4 e to the power of 2 y end exponent plus 5 close parentheses space d y

  • integral subscript 0 superscript ln space 2 end superscript open parentheses 4 e to the power of 2 y end exponent plus 5 close parentheses space d y

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1
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Let R be the region bounded by the graph of x equals ln y, the vertical line x equals 1, and the horizontal lines y equals e and y equals 5. Which of the following gives the area of R?

  • integral subscript 1 superscript ln 5 end superscript e to the power of x space italic d x

  • integral subscript e superscript 5 ln y space italic d y

  • integral subscript 1 superscript ln 5 end superscript open parentheses e to the power of x minus 5 close parentheses space italic d x

  • integral subscript e superscript 5 open parentheses ln y minus 1 close parentheses space italic d y

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2
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What is the area of the region between the graphs of y equals square root of x and y equals negative x from x equals 0 to x equals 4?

  • 8 over 3

  • 32 over 3

  • 40 over 3

  • 24

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3
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Graph of functions y=f(x) and y=h(x) with shaded region between them from x=a to x=c, intersecting the x-axis at x=0 and x=b.

Which of the following integrals represents the area of the shaded region in the figure above?

  • integral subscript a superscript c f open parentheses x close parentheses space d x plus open vertical bar integral subscript 0 superscript b h open parentheses x close parentheses space d x close vertical bar

  • integral subscript a superscript c open parentheses f open parentheses x close parentheses minus h open parentheses x close parentheses close parentheses space d x

  • integral subscript a superscript 0 open parentheses f open parentheses x close parentheses minus h open parentheses x close parentheses close parentheses space d x plus integral subscript 0 superscript b open parentheses h open parentheses x close parentheses minus f open parentheses x close parentheses close parentheses space d x plus integral subscript b superscript c open parentheses f open parentheses x close parentheses minus h open parentheses x close parentheses close parentheses space d x

  • integral subscript a superscript c f open parentheses x close parentheses space d x minus integral subscript a superscript c open vertical bar h open parentheses x close parentheses close vertical bar space d x

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4
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What is the total area enclosed between the curve y equals x cubed minus 8 x squared plus 21 x minus 13 and the line y equals 2 x minus 1?

  • 37 over 12

  • 9 over 4

  • 8 over 3

  • negative 5 over 12

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5
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Graph with x-axis and y-axis showing curve x = f(y). Shaded regions P, Q, R between curves from -1 to 2 on y-axis.

The regions P, Q and R are bounded by the curve x equals f open parentheses y close parentheses and the y-axis. If the area of each region is 6, what is the value of integral subscript negative 1 end subscript superscript 2 open parentheses f open parentheses y close parentheses plus 5 close parentheses space d y?

  • -6

  • -1

  • 9

  • 21

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