Free response with a graphing calculator (Part A)

Area and volume of a region

Unit 8
9 points
About 15 minutes

You can use a calculator on this question, just like on exam day.

A multi-part problem, usually set in a real-world context, where you need a graphing calculator for things like definite integrals, derivatives at a point and solving equations. You show your setup, give decimal answers to three places, and explain or justify your conclusions. On the exam: 2 questions in Part A of the free-response section (30 minutes, graphing calculator required). The whole free-response section is 6 questions in 90 minutes and counts for 50% of the score. AB and BC use the same format; on BC, one of these is usually a parametric, polar or vector question.

The question

Let f(x) = 4 ln(x + 1) and g(x) = x²/2 − x. Let R be the region enclosed by the graphs of f and g. The graphs intersect at x = 0 and at one positive value x = B. For 0 < x < B, the graph of f lies above the graph of g.

Suggested time: 15 minutes

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Part (a)

2 points

Find the area of R.

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Part (b)

3 points

Find the volume of the solid generated when R is revolved about the horizontal line y = −1.

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Part (c)

2 points

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

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Part (d)

2 points

The vertical line x = k divides R into two regions with equal areas. Write, but do not solve, an equation involving one or more integrals whose solution gives k.

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