Lesson 7 of 7 · 11 min

Chapter review

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13 facts

1Area under y = f(x) ≥ 0 from x = a to x = b: A = ∫ₐᵇ f(x) dx.
2Area against the y-axis: A = ∫꜀ᵈ g(y) dy for the curve x = g(y) ≥ 0 between y = c and y = d.
3The elementary strip is dA = y dx (vertical) or dA = x dy (horizontal).
4A part of the curve below the x-axis gives a negative integral; its area is the absolute value.
5Split the interval at every point where the curve crosses the axis, then add the absolute values.
6∫₀^(2π) cos x dx = 0, but the area between y = cos x and the x-axis on [0, 2π] is 4.
7Area of the circle x² + y² = a² is πa².
8Area of the ellipse x²/a² + y²/b² = 1 is πab.
9Use symmetry: find the area of one quarter or one half and multiply.
10The region between y = x² and y = 4 has area 32/3, which is 2/3 of the enclosing rectangle.
11Area between y² = 4ax and its latus rectum is 8a²/3.
12Between two curves (JEE): A = ∫ₐᵇ (upper − lower) dx, with a and b where the curves meet.
13Area between y = x and y = x² is 1/6; between y² = 4x and x² = 4y it is 16/3.

Common traps

Integrating straight across a point where the curve crosses the x-axis.

Find the zeros inside [a, b], split there and add the absolute values of the pieces.

Reporting a negative number as an area.

Area is always positive; take |∫f dx| for a piece below the axis.

Taking y = ±√(a² − x²) and integrating both signs over the whole circle, which cancels to 0.

Work in the first quadrant, where y ≥ 0, and multiply by 4.

Using πa² for an ellipse or πab with the wrong semi-axes.

Read a and b from x²/a² + y²/b² = 1: a is the square root of the number under x².

Using vertical strips where the top boundary changes formula, then missing a piece.

Either split at the change or switch to horizontal strips.

Guessing the limits of an area between curves.

Solve f(x) = g(x) for the intersection points and use them as limits.

Writing ∫(lower − upper) and getting a negative answer.

Check which curve is on top at a sample point inside the interval.

7 to know

Vertical strips

A = ∫ₐᵇ y dx = ∫ₐᵇ f(x) dx

f ≥ 0 on [a, b].

Horizontal strips

A = ∫꜀ᵈ x dy = ∫꜀ᵈ g(y) dy

g ≥ 0 on [c, d].

Below the axis

A = |∫ₐᵇ f(x) dx| when f ≤ 0 on [a, b]

Split at every zero first.

Circle

Area of x² + y² = a² is πa²

4 × first-quadrant area.

Ellipse

Area of x²/a² + y²/b² = 1 is πab

Circle stretched by b/a.

Parabola and latus rectum

Area between y² = 4ax and x = a is 8a²/3

Double the upper half.

Between curves (JEE)

A = ∫ₐᵇ [f(x) − g(x)] dx

f ≥ g on [a, b]; a, b from f = g.

5 terms

Elementary area: The area dA of one thin strip, y dx or x dy, standing for every strip in the region.
Vertical strip: A thin slice parallel to the y-axis, of height y and width dx.
Horizontal strip: A thin slice parallel to the x-axis, of length x and thickness dy.
Ordinate: A vertical line x = a used as a side of the region.
Latus rectum: The chord of a parabola through its focus, perpendicular to its axis; x = a for y² = 4ax.