Given are the two curves y = f(x) and y = g(x). equating both the curves to get the value of interval [a,b] where the given curves intersects.
Step 2 :
The area between the two curves bounded by the x axis is given by the formula
A = $\int_{a}^{b}$[f(x) − g(x)]dx where f(x) is upper curve and g(x) is the lower curve.
Step 3 :
Simplify the problem by substituting the value of the limit to get the answer.
Find the area bounded by the curve y = 6x + x3 below by y = 5x2.
Step 1 :Find the intersection points by equalizing the two curves: 6x + x3 = 5x2.
6 + x2 = 5x.
x2 - 5x + 6 = 0
x2 - 3x - 2x + 6 = 0
x(x-3) -2 (x-3) = 0
The two points where the curves intersects are x=2 and x=3.
Step 2 :The difference between these curves at intersection points are given by:
A = $\int_{2}^{3}$ $[(6x + x^{3}) - 5x^{2}]$ dx
= $\int_{2}^{3}$ $[ x^{3} - 5x^{2} + \frac{x^{2}}{2}]$ dx
= $[\frac{x^{4}}{4} - 5 \frac{x^{3}}{3} + 3 \frac{x^{2}}{2}]^{3}_{2}$
= [$\frac{3^{4}}{4}$ - 5 $\frac{3^{3}}{3}$ + 3 (3)2] - [$\frac{2^{4}}{4}$ - 5 $\frac{2^{3}}{3}$ + 3(2)2].
Step 3 :A = [$\frac{81 \times 3 + 54 \times 6 - 135 \times 4 - 16 \times 3 - 24 \times 6 + 40 \times 4 }{12}$]
= - $\frac{5}{12}$.
Answer :The area between two curves is - $\frac{5}{12}$.
Find the area between the curves y = 2 - x2 and y = x.
Step 1 :given curves are y = 2 - x2 and y = x.
equating both the curves we get,
- x2 + x + 2 = 0
- x2 + 2x - x + 2 = 0
-x(x - 2) - (x - 2) = 0
(-x - 1)(x - 2) = 0
x = -1, 2 are the two intersecting points.
Step 2 :The difference between these curves at intersection points are given by:
A = $\int_{-1}^{2} [(2 - x^{2}) - x]$ dx
= $\left[ 2x - \frac{x^{3}}{3} - \frac{x^{2}}{2} \right]_{1}^{-2}$
= $\left[ 2(1) - \frac{1^{3}}{3} - \frac{1^{2}}{2} \right]_{1}^{-2} - \left[ 2(-2) - \frac{(-2)^{3}}{3} - \frac{(-2)^{2}}{2} \right]$
Step 3 :A = $\left[2 - \frac{1}{3} - \frac{1}{2} + 4 - \frac{8}{3} + \frac{4}{2} \right]$
= $\frac{9}{2}$
Answer :The Area between two curves is $\frac{9}{2}$.