Areas and Volumes Questions

The types of areas and volumes calculations problems discussed here have been frequently been used to test AP Calculus AB exam takers.

Examples.

1. The most common type of area calculation is the area between two functions in the x, y plane.

In the figure above, the area A between the two functions is:
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If the problem is part of the exam section that allows the use of a calculator, you may evaluate the integral by using the graphic calculator, otherwise you have to calculate it.
In most of the cases you will need to calculate the points of intersection of the two graphs, a and b, by solving the equation f(x) = g(x).
In some cases the area that needs to be calculated is delimited by more than two functions.

2. Another frequent type of problem is to calculate the volum of a solid with a base defined by functions or geometric shapes. The cross sections perpendicular to the x axis of the solid are either geometrical shapes, of specific functions.

If A(x) is the area of the cross sections as a function of x, the volume of the solid is:
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For a square cross section, the volume is:
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For a semi circular cross section, the volume is:
ap, calculus, area,

For a cross section specified by the function h(x), the volume is:
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3. Volume of the solid generated when region S is revolved about the line y = c.
If g(x) is the outer radius and g(x) is the inner radius,
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If g(x) is the inner radius and g(x) is the outer radius,
area, volume, graph

4. Volume of the solid generated when region S is revolved about the line x = d.
To solve such a problem you need to write the two functions as x = i(y) and x = j(y).
If h(x) is the outer radius and i(x) is the inner radius,
ap, calculus, area, volume

If h(x) is the inner radius and i(x) is the outer radius,
ap, calculus, area, volume, graph

Examples of areas and volumes calculations

1. Let f and g be the functions given by:

Let R be the region in the first and second quadrants enclosed by the graphs of f and g. Find the area of R.

Solution: The answer can be found numerically by integrating f(x) - g(x) between their points of intersection. First we need to find these points:

4 - x² = .25x² + x + 1
1.25² + x - 3 = 0 with the solutions x = -2 and x = 1.2

2. Let f and g be the functions given by:

Let R be the region in the first and second quadrants enclosed by the graphs of f and g. Find the volume of the solid generated when R is revolved about the x axis.

Solution:

3. Let R be the region in the first quadrant bounded by the y axis and the graphs of f(x) = 3 + ln(x + 1) and g(x) = 2^3x. Find the area of R.

Solution:
3 + ln(x + 1) = 2^3x solved numerically gives x = .598.

4. Let R be the region in the first quadrant bounded by the graphs of f(x) = 3 + ln(x + 1) and g(x) = 2^3x. 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 this solid.

Solution:
3 + ln(x + 1) = 2^3x solved numerically gives x = .598.