Question

Find the area of the region R bounded by the line y = x + 2 and the parabola y = x
^2 − 4

Answer 1

To find the area of the region R bounded by the line y = x + 2 and the parabola y = x^2 − 4, we need to determine the x-values where the two curves intersect. So, the area of the region R is 18 square units.

Step-by-step explanation:

To find the area of the region R bounded by the line y = x + 2 and the parabola y = x^2 − 4, we need to determine the x-values where the two curves intersect. We can do this by setting the two equations equal to each other:

x + 2 = x^2 − 4

Rearranging the equation to standard form:

x^2 − x − 6 = 0

This quadratic equation can be factored as:

(x - 3)(x + 2) = 0

Therefore, the x-values where the two curves intersect are x = 3 and x = -2. To find the area of the region, we integrate the difference between the two curves:

Area = ∫(x + 2 - (x^2 − 4)) dx, from x = -2 to x = 3

After evaluating the integral, the area of the region R is 18 square units.