Study GuidesExam QuestionsPast Papers
AP®MathsCollege BoardCalculus ABExam QuestionsUnit 8: Applications of IntegrationAreas

Areas (College Board AP® Calculus AB)

Exam Questions

1 hour30 questions

Select a question set to download for
Areas

Select an answer set to view for
Areas

Easy
Medium
Hard
1
Sme Calculator3 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.

How did you do?

Did this page help you?

2
Sme Calculator3 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

How did you do?

Did this page help you?

3
Sme Calculator3 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.

How did you do?

Did this page help you?

4
Sme Calculator3 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.

How did you do?

Did this page help you?

5
Sme Calculator3 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.

How did you do?

Did this page help you?

1
Sme Calculator3 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.

How did you do?

Did this page help you?

2
Sme Calculator3 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.

How did you do?

Did this page help you?

3
Sme Calculator4 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.

How did you do?

Did this page help you?

4
Sme Calculator4 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

How did you do?

Did this page help you?

5
Sme Calculator4 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

How did you do?

Did this page help you?

1
Sme Calculator4 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.

How did you do?

Did this page help you?

2
Sme Calculator4 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.

How did you do?

Did this page help you?

3
Sme Calculator6 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.

How did you do?

Did this page help you?

4
Sme Calculator6 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

How did you do?

Did this page help you?

5
Sme Calculator6 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

How did you do?

Did this page help you?