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AP Calculus AB: 10.3.1 Finding Areas by Integrating with Respect to y: Part One

Mathematics8 CardsCreated 3 months ago

This section introduces the idea of integrating with respect to y by stacking horizontal rectangles, which can be more effective when curves are not easily expressed as functions of x. It helps determine when to use x- or y-integration based on which direction defines the region more clearly.

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Finding Areas by Integrating with Respect to y: Part One

Sometimes it is easier to stack horizontal rectangles instead of vertical rectangles when finding the area bounded by two curves

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Term
Definition

Finding Areas by Integrating with Respect to y: Part One

Sometimes it is easier to stack horizontal rectangles instead of vertical rectangles when finding the area bounded by two curves

note

  • You have already seen how to find the area between two curves by integrating with respect to x. Integrating with respect to x stacks an inf...

Is the area bound by these curves x‑easy, y‑easy, both, or neither?

Neither x‑easy nor y‑easy

When integrating with respect to y, what is different about the boundary points (the boundaries of integration)?

The boundaries use the y-value of the point in question instead of the x-value.

Is the area bound by these curves x‑easy, y‑easy, both, or neither?

Both x‑easy and y‑easy

Is the area bound by P (x) and g (x) x‑easy, y‑easy, both, or neither?

y‑easy

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AP Calculus AB: 10.3.1 Finding Areas by Integrating with Respect to y: Part One
TermDefinition

Finding Areas by Integrating with Respect to y: Part One

Sometimes it is easier to stack horizontal rectangles instead of vertical rectangles when finding the area bounded by two curves

note

  • You have already seen how to find the area between two curves by integrating with respect to x. Integrating with respect to x stacks an infinite number of vertical rectangles side by side and adds together their areas.

  • But sometimes curves are not well-defined for integrating with respect to x. Notice that you have to change how you define the height of the rectangles bound by these curves. Watch for cases where the curves are not functions of x.

  • If the heights of the rectangles are well-defined using vertical rectangles, integrate with respect to x.

  • You can stack horizontal rectangles as well. If the heights of the rectangles are well-defined using horizontal rectangles, integrate with respect to y.

Is the area bound by these curves x‑easy, y‑easy, both, or neither?

Neither x‑easy nor y‑easy

When integrating with respect to y, what is different about the boundary points (the boundaries of integration)?

The boundaries use the y-value of the point in question instead of the x-value.

Is the area bound by these curves x‑easy, y‑easy, both, or neither?

Both x‑easy and y‑easy

Is the area bound by P (x) and g (x) x‑easy, y‑easy, both, or neither?

y‑easy

Is the area bound by these curves x‑easy, y‑easy, both, or neither?

x‑easy

Which of the following best explains the concept of integrating with respect to y ?

Integrating with respect to y is like integrating with respect to x, but instead of the internal rectangles being vertical, they are horizontal.