## Applications of the Integrals

# Area between Two Curves

- Area between the two parabolas y
^{2}= 4ax and x^{2}= 4by is = \frac{16\ ab}{3} sq.units - The area region bounded by y
^{2}= 4ax and y = mx is = \frac{8\ a^{2}}{3 m^{3}} sq.units - The area region bounded by y
^{2}= 4ax and its latus rectum x = a is \frac{8\ a^{2}}{3} sq.units - The area region bounded by y
^{2}= 4ax and the line x = b is \frac{8\ a^{\frac{1}{2}}b^{\frac{3}{2}}}{3} sq.units - The area region bounded by x
^{2}= 4by and the line y = a is \frac{8\ b^{\frac{1}{2}}a^{\frac{3}{2}}}{3} sq.units - The area region bounded by x
^{2}= 4ay and y = mx is \frac{8}{3}a^{2}m^{3} sq.units - The area region bounded by y
^{2}= 4ax and the lines x = b, x = c is \frac{8}{3}a^{\frac{1}{2}}\left(c^{\frac{3}{2}}-b^{\frac{3}{2}}\right) sq.units. - The area region bounded by y = ax
^{2}+ bx + c and y = mx + k is \frac{\Delta^{\frac{3}{2}}}{6a^{2}}, where Δ is the discriminent of ax^{2}+ (b − m) x + (c − k) - The area region bounded by x = ay
^{2}+ by + c and x = my + k is \frac{\Delta^{\frac{3}{2}}}{6a^{2}}, where Δ is the discriminent of ay^{2}+ (b − m) y + (c − k) - The area of the parallelogram formed by the lines a
_{1}x + b_{1}y + c_{1}= 0, a_{1}x + b_{1}y + d_{1}= 0 & a_{2}x + b_{2}y + c_{2}= 0, a_{2}x + b_{2}y + d_{2}= 0 is \begin{vmatrix}\frac{(c_{1}-d_{1})(c_{2}-d_{2})}{a_{1}b_{2}-a_{2}b_{1}}\end{vmatrix} sq.units. - The area of the rhombus formed by ax ± by ± c = 0 is \frac{2c^{2}}{|ab|} sq.units
- The area of the triangle by y = |ax + b| with y-axis (or) x = |ay + b| with x-axis is \frac{b^{2}}{a}.
- The area of the triangle formed by the tangent and normal at P(x
_{1}, y_{1}) and x-axis is = \frac{y_{1}^{2}}{2}\begin{vmatrix}M+\frac{1}{M}\end{vmatrix} Where M is slope of tangent at P - The area of the triangle formed by the tangent and normal at P(x
_{1}, y_{1}) with y-axis is = \frac{x_{1}^{2}}{2}\begin{vmatrix}M+\frac{1}{M}\end{vmatrix} Where M is slope of tangent at P

### View the Topic in this video From 12:52 To 53:08

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1. The area of the region enclosed between two curves y = f(x), y = g(x) and the lines x = a, x = b is given by the formula,

Area =\int_{a}^{b} [f(x)-g(x)] \ dx, where, f(x) ≥ g(x) in [a, b]

2. If f(x) ≥ g(x) in [a, c] and f(x) ≤ g(x) in [c, b], a < c < b, then

Area = \int_{a}^{c} [f(x)-g(x)] \ dx+\int_{c}^{b} [g(x)-f(x)] \ dx.