48.4k views
2 votes
How many definite integrals would be required to represent the area of the region enclosed by the curves and , assuming you could not use the absolute value function?

User Kaosad
by
8.9k points

1 Answer

1 vote
The curves y = x and y = 1 - x^2 intersect at the points (-1, 0), (0, 0), and (1, 0). The region enclosed by these curves is a triangle with vertices at these points.

To find the area of this region without using the absolute value function, we can split the triangle into two parts: the part above the x-axis and the part below the x-axis.

For the part above the x-axis, the bounds of integration are x = 0 to x = 1. The integrand is y = x, the equation of the upper curve. The integral is:

∫[0,1] x dx = 1/2

For the part below the x-axis, the bounds of integration are x = -1 to x = 0. The integrand is y = 1 - x^2, the equation of the lower curve. However, since we cannot use the absolute value function, we need to split this integral into two parts as well. When x is between -1 and 0, the lower curve is y = 1 - x^2, and when x is between 0 and 1, the lower curve is y = x. Therefore, we have:

∫[-1,0] (1 - x^2) dx + ∫[0,1] x dx

Evaluating the first integral gives:

∫[-1,0] (1 - x^2) dx = x - (x^3/3) evaluated from -1 to 0 = 1/3

Therefore, the area of the region enclosed by the curves is:

1/2 + 1/3 = 5/6

So, the area of the region can be represented using two definite integrals.
User Gorpacrate
by
8.3k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories