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AP Calculus AB: 10.2.2 Limits of Integration and Area

Mathematics7 CardsCreated 3 months ago

This section focuses on determining the area between curves when the limits of integration are not explicitly given. It emphasizes finding points of intersection algebraically to set the correct bounds, setting up the definite integral using the upper minus lower function, and carefully evaluating to match the visual region.

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Limits of Integration and Area

To find the area of a region bounded by the graphs of two functions, find the limits of integration by determining where the graphs intersect. Then take the definite integral of the difference of the two functions along that interval.

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Key Terms

Term
Definition

Limits of Integration and Area

To find the area of a region bounded by the graphs of two functions, find the limits of integration by determining where the graphs intersect. Then...

note

  • You will have to do some algebra if the endpoints of the region are not stated.

  • Remember that the two curves cross when the ...

Find the area of the region bound by y = x + 5, y = −x/2 + 1, and the y-axis.

16/3

Find the area of the region bound by y = cos x, y = −sin x, and the y-axis in the second quadrant.

√2−1

Find the area of the region bound by y = x/2 and y=√x

4/3

What is the area bound between the curves f (x) = x ^2 and g (x) = x ^3?

A = 1/12

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AP Calculus AB: 10.2.2 Limits of Integration and Area
TermDefinition

Limits of Integration and Area

To find the area of a region bounded by the graphs of two functions, find the limits of integration by determining where the graphs intersect. Then take the definite integral of the difference of the two functions along that interval.

note

  • You will have to do some algebra if the endpoints of the region are not stated.

  • Remember that the two curves cross when the two functions are equal. So set their expressions equal to each other and solve for the endpoints.

  • Once you have the endpoints, you can set up the definite integral normally.

  • Remember to take the upper curve and subtract the lower curve.

  • Be careful! It is very easy to make an algebraic or arithmetic mistake while evaluating definite integrals. Always double check that your answer agrees with your sketch of the region.

Find the area of the region bound by y = x + 5, y = −x/2 + 1, and the y-axis.

16/3

Find the area of the region bound by y = cos x, y = −sin x, and the y-axis in the second quadrant.

√2−1

Find the area of the region bound by y = x/2 and y=√x

4/3

What is the area bound between the curves f (x) = x ^2 and g (x) = x ^3?

A = 1/12

Find the area of the region bounded by y=1/x, x=1, x=2, and y=−1.

1 + ln 2