# How do you sketch the region enclosed by #y=x+1, y=9-x^2, x=-1, x=2# and find the area?

If we look at the two functions in the given interval we see that:

The area between these two functions in the given interval will be:

Plugging in upper and lowers bounds:

GRAPH:

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To sketch the region enclosed by the given equations and find the area, follow these steps:

- Plot the graphs of the equations ( y = x + 1 ) and ( y = 9 - x^2 ) on the same set of axes.
- Identify the points of intersection of the two curves.
- Determine the x-values where the enclosed region begins and ends by analyzing the given bounds: ( x = -1 ) and ( x = 2 ).
- Calculate the area of the enclosed region using definite integrals, integrating with respect to x from the lower bound to the upper bound. [ A = \int_{-1}^{2} (9 - x^2 - (x + 1)) , dx ]

After evaluating the integral, you will find the area of the enclosed region.

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