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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 8: Applications of IntegrationAreas

Areas (College Board AP® Calculus AB)

Exam Questions

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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.

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2
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.

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3
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.

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4
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.

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5
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.

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