Application of Integrals — Formula Sheet

dA = y dx

A = ∫_a^b dA = ∫_a^b y dx = ∫_a^b f(x) dx

A = ∫_c^d x dy = ∫_c^d g(y) dy

|∫_a^b f(x) dx|

A = |A1| + A2

The circle x² + y² = a² is symmetrical about both the x-axis and y-axis.

The ellipse x²/a² + y²/b² = 1 is also symmetrical about both axes.

For the line y = 3x + 2, the x-axis is met at x = -2/3.

The area bounded by y = 3x + 2, x-axis, x = -1 and x = 1 is 13/3.

The curve y = cos x between x = 0 and x = 2π has total area 4.

A = \int_a^b f(x) dx gives the area under y = f(x) between x = a and x = b.

A = \int_c^d g(y) dy gives the area when the curve is written as x = g(y).

The circle x^2 + y^2 = a^2 is symmetric about both axes.

Total area = 4 times the first-quadrant area.

The area bounded by the curve y = f(x), the x-axis, and the ordinates x = a and x = b is A = \int_a^b f(x) dx.

The area bounded by the curve x = g(y), the y-axis, and the lines y = c and y = d is A = \int_c^d g(y) dy.