https://youtu.be/TjF-ineixvQ

In this video I go over a useful example on finding the area between a curve by switching the variables to be functions of y as opposed to x. The example is finding the area enclosed by the line y = x + 1 and the parabola y<sup>2</sup> = 2x + 6. Writing the functions in terms of y makes it easier to solve the integral without having to break up the function into 2 regions. This is a very important example so make sure to watch it!

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# Integrals and Areas Between Curves: Examples Part 5

![Integrals and Areas Between Curves Examples 5.jpeg](https://cdn.steemitimages.com/DQmRihb4qn4oSvXv7BseJouTLXGZ7ZMmGzj41crdqgR3mH8/Integrals%20and%20Areas%20Between%20Curves%20Examples%205.jpeg)

**Example:**

Find the area enclosed by the line y = x - 1 and the parabola y<sup>2</sup> = 2x +6

**Solution:**

![image.png](https://cdn.steemitimages.com/DQmaK2rm8QQZW5g1Hg4TCLg7aypXXcrrUtQRahcckAHytM6/image.png)

![image.png](https://cdn.steemitimages.com/DQmUrWgfXdw5AtYkoqpRrN5boCnRxBCTEbRGHhpxzH9Hgvt/image.png)

![image.png](https://cdn.steemitimages.com/DQmT5YfAfDUWrGSmYarzdJhFbLGcFoA87T3wF4cEXoJLwsS/image.png)

If we stuck with using functions of x, then we would need to solve 2 integrals by breaking up y into 2 functions.


![image.png](https://cdn.steemitimages.com/DQme5zJbGXtsDdK4tDSULkVgECVnpcoi3uovEcEQaEm1g5m/image.png)