Volumes with Cross Sections (College Board AP® Calculus BC): Exam Questions

49 mins23 questions
1
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2 marks
Graph showing the shaded region R bounded by the y-axis, the line x=3, and the curves y=e^cosx and y=sqrt(x)-3

Let R be the region enclosed by the graphs of f open parentheses x close parentheses equals e to the power of cos x end exponent and g open parentheses x close parentheses equals square root of x minus 3, the y-axis, and the vertical line x equals 3, as shown in the figure above.

Region R is the base of a solid. For the solid, at each x the cross section perpendicular to the x-axis has area A open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 5 end fraction. Find the volume of the solid.

2
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3 marks

Let f open parentheses x close parentheses equals 1 half x squared minus 3 x plus 4 and g open parentheses x close parentheses equals 4 cos open parentheses pi over 8 x close parentheses. Let R be the region bounded by the graphs of f and g, as shown in the figure.

The region R is the base of a solid. For this solid, each cross-section perpendicular to the x-axis is a square. Write down an integral expression that gives the volume of the solid and find its value using your calculator.

Graph showing two intersecting curves, f and g. Region R is between the curves. Curves intersect each other on the y axis and on the x axis.
3
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4 marks

Let R be the region in the first and second quadrants bounded above by the graph of y equals fraction numerator 40 over denominator 1 plus x squared end fraction and below by y equals 4.

Graph showing a symmetrical bell shaped curve centred on the y axis. A horizontal line at a positive y value intersects it in two places

The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semicircles. Find the volume of this solid.

1
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3 marks
A graph showing the region enclosed by the two curves f(x)=ln(sqrt(2-x)) and g(x)=x^4+2x^3-2x-1

Let f and g be functions defined by f open parentheses x close parentheses equals ln square root of 2 minus x end root and g open parentheses x close parentheses equals x to the power of 4 plus 2 x cubed minus 2 x minus 1. The graphs of f and g, shown in the figure above, intersect at x equals P and x equals 1, where P less than 0.

The region enclosed by the graphs of f and g is the base of a solid. Cross sections of the solid taken perpendicular to the x-axis are squares. Find the volume of the solid.

2
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4 marks
Graph showing the shaded region R bounded by the x-axis, the line x=5, and the curve y=sqrt(2x-1)

Let R be the region bounded by the x-axis, the line x equals 5, and the graph of y equals square root of 2 x minus 1 end root, as shown in the figure above.

Region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is a semicircle. Find the volume of the solid.

3
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3 marks

Let R be the region in the first quadrant bounded by the x-axis and the graphs of y equals ln space open parentheses x plus 1 close parentheses and y equals 6 minus 2 x, as shown in the figure below.

Region R is the base of a solid. For the solid, at each x the cross section perpendicular to the x-axis is a semicircle. Find the volume of the solid.

Graph with a curve intersecting a straight line and axes. The region between them and the x-axis is labelled R.
4
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2 marks

Let R be the region in the first quadrant bounded by the graph of y equals 3 square root of x​, the horizontal line y equals 9, and the y-axis.

The region R is the base of a solid. For each y, where 0 less or equal than y less or equal than 9, the cross-section of the solid taken perpendicular to the y-axis is a rectangle whose height is 4 times the length of its base in region R. Write an integral expression that gives the volume of the solid.

Graph of a curve passing through the origin, intersecting a horizontal line at y=9, with a point labelled (9,9). Region R is formed between the curve, the line, and the y-axis
5
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3 marks

Let R be the region enclosed by the graph of f open parentheses x close parentheses equals negative x squared open parentheses x minus 3 close parentheses squared plus 2 and the horizontal line y equals 2, as shown in the figure below.

Graph of a quartic curve with an m shape which intersects a horizontal line at the local maximum points, forming a region R between the curve and line

Region R is the base of a solid. For this solid, each cross-section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid.

1
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4 marks
Graph showing the shaded region R enclosed by the x-axis, the curve y=sqrt(x-1) and the line y=10-2x

Let R be the region in the first quadrant bounded by the x-axis and the graphs of y equals square root of x minus 1 end root and y equals 10 minus 2 x, as shown in the figure above.

Region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is a rectangle, with its height equal to one third of its base. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.

2
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3 marks
Graph showing the shaded region R bounded by the graphs of f(x)=x^2-2x+3 and g(x)=x^3-3x^2+x+3

Let f and g be functions defined by f open parentheses x close parentheses equals x squared minus 2 x plus 3 and g open parentheses x close parentheses equals x cubed minus 3 x squared plus x plus 3. The region R is the region enclosed by the graphs of f and g between x equals a and x equals b where a comma b greater than 0, as shown in the figure above.

The region R is the base of a solid. Cross sections of the solid taken perpendicular to the x-axis are equilateral triangles. Write, but do not evaluate, an integral that gives the volume of the solid.

3a
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3 marks

Let R be the region in the first quadrant enclosed by the graphs of y equals 3 x and y equals x squared plus x, as shown in the figure.

Graph depicting a curve and a line intersecting at point (2, 6) and the origin. The region formed is labeled as R

The region R is the base of a solid. For this solid, at each x, the cross-section perpendicular to the x-axis has an area given by A open parentheses x close parentheses equals sin open parentheses pi over 2 x close parentheses. Find the volume of the solid.

3b
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3 marks

Another solid has the same base R. For this solid, the cross-sections perpendicular to the y-axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid.