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
Sme Calculator
3 marks

Find the finite area bounded by the y-axis and the graph of x equals open parentheses y minus 2 close parentheses open parentheses y plus 1 close parentheses.

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2
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3 marks

Let S be the region enclosed by the graphs of p open parentheses x close parentheses equals 3 minus x squared andq open parentheses x close parentheses equals negative x cubed plus 2 x, the y-axis, and the vertical line x equals 2, as shown in the figure below.

Find the area of S.

Graph of two curves, one above the other, and a vertical line, forming an area

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3
Sme Calculator
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.

Find the area of region R.

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

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4
Sme Calculator
3 marks

Let R be the shaded region bounded by the graphs of y equals square root of 2 x end root, y equals e to the power of negative 2 x end exponent and the vertical line x equals 1, as shown in the figure below.

Find the area of R.

Graph with two intersecting curves on an axis, and a vertical line which intersects both. One curve slopes downward; the other slopes upward. The region in the middle is labelled R.

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5
Sme Calculator
3 marks

Find the exact total area enclosed by the graph of y equals x open parentheses x minus 2 close parentheses, the line x equals 3, and the x-axis.

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6
Sme Calculator
2 marks

What information does integral subscript 0 superscript 10 square root of 1 plus open parentheses f apostrophe open parentheses x close parentheses close parentheses squared end root space d x provide about the graph of the function f?

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7
Sme Calculator
2 marks

Figures 1 and 2 illustrate regions in the first quadrant associated with the graphs of y = \frac{1}{x} and y = \frac{1}{x^{2}}, respectively. In Figure 1, let R be the region bounded by the graph of y = \frac{1}{x}, the x-axis, and the vertical lines x = 1 and x = 5. In Figure 2, let W be the unbounded region between the graph of y = \frac{1}{x^{2}} and the x-axis that lies to the right of the vertical line x = 3.

Figure 1 shows region R under y = 1/x from x = 1 to x = 5 in the first quadrant; Figure 2 shows the unbounded region W under y = 1/x^2 to the right of x = 3 in the first quadrant

Find the area of region R.

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1
Sme Calculator
3 marks
Graph showing two intersecting curves

Let f and g be the functions defined by f open parentheses x close parentheses equals 5 ln open parentheses 2 x plus 9 close parentheses and g open parentheses x close parentheses equals x to the power of 6 plus 4 x cubed. The graphs of f and g, shown in the figure above, intersect at x equals A and x equals B where A less than 0 and B greater than 0.

Find the area of the region enclosed by the graphs of f and g.

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2
Sme Calculator
3 marks
A curve on an x-y graph, starting at the origin, rising to a peak, and descending back to the x-axis, forming an arch shape.

The graph of the family of functions y equals a x square root of 9 minus x squared end root for x greater or equal than 0, where a is a positive constant, has the general shape shown in the figure above.

Find the area of the region in the first quadrant bounded by the x-axis and the graph of y equals a x square root of 9 minus x squared end root for a equals 8.

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3
Sme Calculator
4 marks
Graph of curves x+y²=1 and x+2y²=2 intersecting, forming a shaded region labeled R. Axes are marked x and y.

The region R is bounded by the curves with equations x plus y squared equals 1 and x plus 2 y squared equals 2. Find the area of R.

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4
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4 marks

Let R and S be the regions bounded by the graphs of y equals negative sin open parentheses pi italic space x close parentheses and y equals x cubed minus 9 x between x equals negative 3 and x equals 3, as shown below.

Find the sum of the areas of the regions R and S, writing your answer in an exact form.

Graph of a cubic and a negative sin curve between -3 and 3, which is symmetrical and has regions R and S formed between the curves

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5
Sme Calculator
4 marks

Function f is given by f open parentheses x close parentheses equals fraction numerator x over denominator x squared plus 4 end fraction. Find the area of the region bounded by f open parentheses x close parentheses, the x-axis, and the vertical lines x equals negative 2 and x equals 4. The graph of f and the the two vertical lines are shown below.

Write your answer in an exact form.

A curve which is below the x axis for negative values and above the x axis for positive values, and vertical lines at x=-2 and x=4 so two areas are formed

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6
Sme Calculator
3 marks

Find the length of the curve y equals f open parentheses x close parentheses on the interval 0 less or equal than x less or equal than 5 where f open parentheses x close parentheses equals 10 cos open parentheses fraction numerator pi x over denominator 5 end fraction close parentheses .

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7
Sme Calculator
2 marks

The shaded region R is bounded by the graphs of the functions f and g, where f \left(x\right) = x^{2} - 2 x and g \left(x\right) = x + \text{sin} \; \left(\pi x\right), as shown in the figure.

Shaded region R between curves y = f(x) and y = g(x) on x–y axes, from x = 0 to x = 3, with point (3, 3) marked on g(x)

(Note: Your calculator should be in radian mode.)

Find the area of R. Show the setup for your calculations.

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8
Sme Calculator
3 marks
Cartesian graph with x and y axes showing a smooth curve dipping below y = −1 near x = −1.5, rising through the origin, and passing a marked point near (1, 1)

Let f and g be the functions defined by f(x) = \ln(x + 3) and g(x) = x^4 + 2x^3. The graphs of f and g, shown in the figure above, intersect at x = -2 and x = B, where B > 0.

Find the area of the region enclosed by the graphs of f and g.

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9
Sme Calculator
2 marks

The functions f and g are defined by f(x) = x^2 + 2 and g(x) = x^2 - 2x, as shown in the graph.

Graph of curves y = f(x) and y = g(x) with shaded regions R (0≤x≤2 under f) and S (2≤x≤5 between g and the x-axis) on x–y axes up to 25.

Let R be the region bounded by the graphs of f and g, from x = 0 to x = 2, as shown in the graph. Write, but do not evaluate, an integral expression that gives the area of region R.

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10
Sme Calculator
3 marks

The graphs of the functions f and g are shown in the figure for 0 \leq x \leq 3. It is known that g(x) = \frac{12}{3 + x} for x \geq 0. The twice-differentiable function f, which is not explicitly given, satisfies f(3) = 2 and \int_{0}^{3} f(x) \, \text{d} x = 10.

The shaded region enclosed by y = f(x) (the upper curve) and y = g(x) (the lower curve) from (0, 4) to (3, 2) in the xy-plane]

Find the area of the shaded region enclosed by the graphs of f and g.

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11a
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2 marks

The function f is twice differentiable for all x with f(0) = 0. Values of f ', the derivative of f, are given in the table for selected values of x.

x

0

\pi

2 \pi

f ' (x)

5

6

0

For x \geq 0, the function h is defined by h(x) = \int_{0}^{x} \sqrt{1 + (f ' (t))^{2}} \, \text{d} t. Find the value of h ' (\pi). Show the work that leads to your answer.

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11b
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2 marks

What information does \int_{0}^{\pi} \sqrt{1 + (f ' (x))^{2}} \, \text{d} x provide about the graph of f?

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1
Sme Calculator
4 marks
Graph showing intersection of two curves, labeled points a, b, c, and regions between the two curves of R and S.

Let f and g be the functions defined by f open parentheses x close parentheses equals 2 plus 1.2 x plus e to the power of x squared minus 1.8 x end exponent and g open parentheses x close parentheses equals x to the power of 4 minus 6 x squared plus 5 x plus 3. Let R and S be the two regions enclosed by the graphs of f and g shown in the figure above. Points a comma space b and c are the intersections of the two graphs.

Find the sum of the areas of regions R and S.

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2
Sme Calculator
4 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 3 x plus 1 close parentheses and y equals 4 minus 2 x, as shown in the figure below.

The horizontal line y equals k divides R into two regions of equal area. Write an equation involving one or more integrals whose solution allows the value of k to be found. You do not need to solve the equation.

Graph with a curve intersecting a straight line and axes. The region between them is labelled R.

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3
Sme Calculator
6 marks

Find the exact area of the region enclosed by the graph of y equals x square root of 1 plus x end root, the x-axis, and the line x equals 1.

Write your answer in the form a open parentheses b plus c to the power of d close parentheses where b and c are positive integers and a and d are positive rational numbers to be found.

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4
Sme Calculator
6 marks

The graph below shows the functions f open parentheses x close parentheses equals 2 cos open parentheses 2 x close parentheses plus 2 and g open parentheses x close parentheses equals negative 2 cos open parentheses 2 x close parentheses.

Find the area of the shaded region.

Graph showing two oscillating curves, f(x) and g(x), intersecting at multiple points. The area between the first negative intersection and the first positive intersection is shaded, as well as between the first and second positive intersections

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5
Sme Calculator
6 marks

The graph below shows the functions f open parentheses x close parentheses equals cos space x and g open parentheses x close parentheses equals sin space 2 x.

One of the points of intersection of f open parentheses x close parentheses and g open parentheses x close parentheses is at x equals pi over 6.

Find the ratio of the areas R subscript 1 colon R subscript 2 in the form a colon b where a and b are integers.

Graph showing curves f(x) and g(x) intersecting, with shaded regions R1 and R2 between them

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6
Sme Calculator
4 marks

Find, in terms of k, the arc length of the curve y equals fraction numerator e to the power of x plus e to the power of negative x end exponent over denominator 2 end fraction on the interval open square brackets negative k comma space k close square brackets where k greater than 0.

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7
Sme Calculator
3 marks
Cartesian axes with origin O and a smooth curve starting at the origin, rising to a maximum, then bending down to meet the positive x-axis again

A company designs spinning toys using the family of functions y = cx\sqrt{4 - x^2}, where c is a positive constant. The figure above shows the region in the first quadrant bounded by the x-axis and the graph of y = cx\sqrt{4 - x^2}, for some c. Each spinning toy is in the shape of the solid generated when such a region is revolved about the x-axis. Both x and y are measured in inches.

Find the area of the region in the first quadrant bounded by the x-axis and the graph of y = cx\sqrt{4 - x^2} for c = 6.

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8
Sme Calculator
4 marks

Let R be the region enclosed by the graphs of g\left(x\right) = -2 + 3\text{cos}\left(\frac{\pi}{2}x\right) and h\left(x\right) = 6 - 2\left(x-1\right)^{2}, the y-axis, and the vertical line x = 2, as shown in the figure below.

Shaded region R on Cartesian axes, bounded by curves between x = 0 and x = 2, crossing the x-axis near x = 1 and extending above and below it

Find the area of R.

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