Concept 09 / 10Integral Calculus
Area Under Curves and Between Curves
Area bounded by a curve and the x-axis as an integral of |f|, areas using horizontal strips, the area between two curves as the integral of (upper minus lower), finding intersection points, and standard areas such as the ellipse and the parabola with its chord.
The definite integral gives signed area, but 'area' in geometry is always positive. So the area between , the -axis and the lines , is
.
In practice, find where changes sign in , split the interval there, and add the absolute values of the pieces. For example, the area between and the axis over is , while .
Area between two curves. If on , the area between them is
,
the integral of the height of a thin vertical strip, 'upper minus lower', as in the figure. This formula works regardless of whether the curves are above or below the axis, because only the difference matters. If the curves cross inside the interval, split at the crossing points and take upper minus lower on each piece. The limits are usually the -coordinates of the intersection points, found by solving .
Horizontal strips. If the region is more naturally described with as a function of (for example, bounded by and a line, or by curves that fail the vertical line test), use horizontal strips: . Choosing the strip direction well can turn two integrals into one.
A systematic method.
1. Sketch the curves, find the intersection points, and shade the region.
2. Decide on vertical or horizontal strips so that the upper and lower (or right and left) boundaries are single formulas across the whole range, if possible.
3. Write the strip length as upper minus lower, integrate over the range, and split wherever the boundary formula changes.
4. Use symmetry to reduce the work: for a region symmetric about an axis, compute one half and double it.
Standard areas worth knowing.
Circle : . Ellipse : (compute ).
Area between the parabola and its latus rectum : .
Area between and : .
Archimedes: the area of a parabolic segment cut off by a chord is of the enclosing rectangle, or of the inscribed triangle with the same base and the vertex where the tangent is parallel to the chord.
Area between and on (): .
Areas involving inverse functions. The graph of is the reflection of the graph of in , so areas are preserved under the reflection. This gives for increasing , a quick way to integrate an inverse function.
Key Takeaways:
- Area : split at sign changes.
- Between curves: ; or .
- Limits come from the intersection points.
- Use symmetry and the better strip direction.
- Ellipse: ; and : ; parabola and latus rectum: .
- Parabolic segment of its bounding rectangle.
- Pitfall: integrating across a crossing point, where the pieces cancel.
Brain-Teaser 1: Find the area bounded by and .
Solution: Intersections: , so or , giving and . In the region between them (above the axis), the upper curve is and the lower is . So . With horizontal strips, , the same, as it must be.
Brain-Teaser 2: Find the area of the region above and below .
Solution: Intersections: gives . Upper minus lower is . By symmetry, . Symmetry about the -axis halves the calculation.
Brain-Teaser 3: Find the area of the region .
Solution: By symmetry, work for and double. There , and gives (the root is not in ). So . The modulus makes the upper boundary a V-shape; splitting by the sign of and using symmetry avoids handling the corner explicitly.
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