143k views
2 votes
Find the area of the region enclosed by the curves between their intersections y=x^2and y=2x The area of the region is_____________ square unit(s).

(Simplify your answer. Type an integer or a fraction.)

User EastOcean
by
7.9k points

1 Answer

3 votes

First, we need to determine the points of intersection of the curves y=x^2 and y=2x.

To find these intersections, we set both equations equal to each other:

x^2 = 2x

Rearranging, we find:

x^2 - 2x = 0

By factoring, we find solutions:

x(x - 2) = 0

Therefore, x = 0 and x = 2 are the intersection points of the given curves.

Now that we have the limits of integration, we can find the area between the curves. The area A of a region bounded by two curves y1 and y2 from x=a to x=b is given by:

A = ∫_a^b |y2 - y1| dx

Here, y1 = x^2 and y2 = 2x. Thus, we have:

A = ∫_0^2 |2x - x^2| dx

Since 2x - x^2 is non-negative in the interval [0,2], we are simply finding an area, and there are no negatives due to symmetry about the y-axis, so we can ignore the absolute values:

A = ∫_0^2 (2x - x^2) dx

This integral can be computed using the power rule, where the integral of x^n dx is (1/(n+1))*x^(n+1) from a to b.

A = 4/3 square units

And therefore, the area of the region enclosed by the curves y=x^2and y=2x is 4/3 square units.

User TerDale
by
7.5k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.