$$\DeclareMathOperator{cosec}{cosec}$$

# Integration

## Furthermore

### Areas between curves

We want to find the area between the curves $$y = x^2 + x - 2$$ and $$y = x + 2$$. First, we need to find the points of intersection between the two curves.

We can do this by setting the two equations equal to each other and solving for $$x$$:

\begin{align*} x^2 + x - 2 &= x + 2 \\ x^2 &= 4 \\ (x - 2)(x + 2) &= 0. \end{align*}

The solutions are $$x = 2$$ and $$x = -2$$, so the curves intersect at these points.

Next, we'll find the area between the curves by integrating the difference between the two functions over the interval from $$-2$$ to $$2$$

\begin{align*} \text{Area} &= \int_{-2}^{2} \left( x + 2 - (x^2 + x - 2) \right) \,dx \\ &= \int_{-2}^{2} \left( -x^2 + 4 \right) \,dx \\ &= \left[ - \frac{x^3}{3} + 4x \right]_{-2}^2\\ &=(-8/3+8)-(8/3-8)\\ &=10 \frac23 \text{ square units} \end{align*}

This Finding Areas Under Curves video is from Revision Village and is aimed at students taking the IB Maths Standard level course

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