Find the area of the region bounded?

Question

Find the area of the region bounded by the parabola y=x^2-3 and the line y=1. 

 

Answer: 32/3

 

Please show all your work. 

Answer 1

Greetings, Sun.

 

Remember that area of a region is just the combined area of a lot of tiny rectangles. Draw one tiny (thin) rectangle that goes from the bottom of the region (the parabola) to the top (the line y=1). What is the area of this rectangle?

 

Area is height times width. The width is just however wide we made the rectangle, this is arbitrary so we just call it Δx. The height is the distance between the parabola and the line. We want to keep the area positive so we subtract the bottom from the top (if we did the other way, we would get a negative answer), or 1-(x² - 3)= 4 - x². So the area of each little rectangle is (4-x²)Δx.

 

We are going to turn this into an integral so we need to find the limits. The rectangles we drew were "vertical" so we need to find limits that are horizontal. You graph this and see that the region goes from x=-2 to x=2. So, when we find the combined sum (sigma) of all the little rectangles across the region from x=-2 to x=2, and then we make Δx as small as we can so that we "actually" calculate the area of infinite rectangles, in calculus we know that this is the same as:

 

The integral of (4-x²)dx from -2 to 2. This is what we need to calculate