Lesson 7 of 7 · 11 min
Chapter review
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Must-know facts
13 facts
- 01Area under y = f(x) ≥ 0 from x = a to x = b: A = ∫ₐᵇ f(x) dx.
- 02Area against the y-axis: A = ∫꜀ᵈ g(y) dy for the curve x = g(y) ≥ 0 between y = c and y = d.
- 03The elementary strip is dA = y dx (vertical) or dA = x dy (horizontal).
- 04A part of the curve below the x-axis gives a negative integral; its area is the absolute value.
- 05Split the interval at every point where the curve crosses the axis, then add the absolute values.
- 06∫₀^(2π) cos x dx = 0, but the area between y = cos x and the x-axis on [0, 2π] is 4.
- 07Area of the circle x² + y² = a² is πa².
- 08Area of the ellipse x²/a² + y²/b² = 1 is πab.
- 09Use 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
Where marks are lost
Formulas
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.
Key terms
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.