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AP Calculus AB: 10.2.4 Regions Bound by Several Curves

Mathematics6 CardsCreated 3 months ago

This section explains how to find the area of regions bounded by more than two curves. It involves identifying points of intersection, breaking the integral into multiple parts based on which curves define the top and bottom, and always subtracting the lower function from the upper. Sketching the region helps ensure accuracy.

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Regions Bound by Several Curves

Sometimes regions can be defined by more than two curves. When finding the area of a region bound by more than two curves, you must break the integral into different pieces wherever the curves bounding the region switch.

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Key Terms

Term
Definition

Regions Bound by Several Curves

Sometimes regions can be defined by more than two curves. When finding the area of a region bound by more than two curves, you must break the integ...

note

  • Regions can be defined in many different ways. Some regions are defined by several different curves. Always sketch the region you are worki...

Find the area of the shaded region bound by y = x^2, y = x + 4, y = −x + 6 and the y-axis.

19/3

What is the area of the region R bounded above by the curve y=2−x^2, and bounded below by the curves y=−x, and y=√x?

AR  = 13/6

Find the area of the shaded region bound by y = (x − 1)^2 − 1, y = (x + 1)^2 − 1, and y = 3

18

Find the area of the region bound by the following curves:

y=√x, y=6−x, y=1.

13/6

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AP Calculus AB: 10.2.4 Regions Bound by Several Curves
TermDefinition

Regions Bound by Several Curves

Sometimes regions can be defined by more than two curves. When finding the area of a region bound by more than two curves, you must break the integral into different pieces wherever the curves bounding the region switch.

note

  • Regions can be defined in many different ways. Some regions are defined by several different curves. Always sketch the region you are working with to identify how the region is defined.

  • Notice that this region is defined by three curves. Also, the way the dimensions of the arbitrary rectangles are defined changes when the curves intersect.

  • You will need to work two integrals to find this area.

  • The area of the region is equal to the area of the first part of the region plus the area of the second.

  • Notice that the definite integral changes for the two different pieces. Always take the upper curve and subtract the lower curve. If the curves change, then you will have to take another integral.

  • Always check that your answer matches the sketch of the region. For example, if you got an answer greater than 4 you would know that you made a mistake somewhere.

Find the area of the shaded region bound by y = x^2, y = x + 4, y = −x + 6 and the y-axis.

19/3

What is the area of the region R bounded above by the curve y=2−x^2, and bounded below by the curves y=−x, and y=√x?

AR  = 13/6

Find the area of the shaded region bound by y = (x − 1)^2 − 1, y = (x + 1)^2 − 1, and y = 3

18

Find the area of the region bound by the following curves:

y=√x, y=6−x, y=1.

13/6