What It Means to Integrate with Respect to y
When you find area with respect to , you imagine the region being divided into thin vertical rectangles. The width of each rectangle is , so the area formula uses .
When you integrate with respect to , the slices are horizontal. Each thin rectangle has a small vertical thickness of , and its horizontal length is determined by the difference between the right-hand x-value and the left-hand x-value.
The most important change is this: for an integral with respect to , you subtract left from right. For an integral with respect to , you subtract bottom from top.
The Basic Formula
Suppose a region is bounded on the left by and on the right by , where both curves are written as functions of .
The formula works because a horizontal slice has length . Multiplying that length by the small thickness gives the approximate area of the slice.
How to Set Up an Area Problem with Respect to y
- Sketch or visualize the region. You need to know what the bounded region looks like before deciding which curves are left and right.
- Find the intersection points. These determine the limits of integration. When integrating with respect to , you need the corresponding -values.
- Rewrite the curves as x-functions of y, if necessary. You want equations of the form .
- Identify the right curve and left curve. For a horizontal slice, the width is .
- Set the limits from bottom to top. The lower intersection gives the lower -value and the upper intersection gives the upper -value.
- Integrate and evaluate. Your final answer should represent the area of the region.
Why the Right-minus-Left Formula Works
Imagine drawing a very thin horizontal rectangle inside the region. Its left endpoint lies on one curve and its right endpoint lies on the other.
If the right boundary is and the left boundary is , then the width of the rectangle is:
The thickness of the rectangle is , so its approximate area is:
Adding infinitely many of these horizontal slices produces the definite integral.
Worked Examples
Find the area of the region bounded by and .
Find the area enclosed by and .
Consider the region bounded by the sideways parabola and the vertical line . Find its area using horizontal slices.
How to Know Whether to Integrate with Respect to x or y
On the AP Calculus exam, you are often allowed to choose whichever orientation makes the problem easier. The best choice is the one that lets you describe the region with the fewest pieces.
Use dx When Vertical Slices Are Convenient
- The curves are naturally written as .
- A vertical slice crosses the region only once.
- The top-minus-bottom expression is simple.
Use dy When Horizontal Slices Are Convenient
- The curves are naturally written as .
- A horizontal slice crosses the region only once.
- The right-minus-left expression is simpler.
- Using avoids splitting the region into multiple integrals.
Common Mistakes
This is the most common mistake. When integrating with respect to , your slices are horizontal, so the relevant distance is horizontal. You need right minus left, not top minus bottom.
If you integrate with respect to , your limits must be -values. Find the intersection points first and identify their lower and upper y-coordinates.
A -integral needs the horizontal boundaries expressed as x-values. If your original equations are written as , you may need to rearrange them before setting up the integral.
Do not assume that the same curve is always on the right. Test a convenient y-value inside the region. The curve producing the larger x-value is the right boundary.
Sometimes one horizontal slice cannot be described by the same right-minus-left expression across the entire region. If the boundaries switch, or the region changes shape, split the integral at the appropriate y-value.
A Quick AP Calculus Checklist
- Draw the region.
- Find the intersection points.
- Identify the lower and upper y-values.
- Rewrite the curves as x-functions of y.
- Identify right and left.
- Write right minus left.
- Integrate with respect to y.
- Check that the final answer is positive.
Practice Problems
Find the area enclosed by and using integration with respect to .
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Find the area enclosed by and using .
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A region is bounded by and . Set up, but do not evaluate, an integral with respect to .
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Explain why an area integral with respect to uses right minus left rather than top minus bottom.
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