Question

find the area of the region between the following curves by integrating with respect to y . if necessary, break the region into subregions first. x = y − y 2 and x = − 3 y 2

Answer 1

The area between the curves is found by setting the curves equal to each other to find the intersection points, which are the limits for the integral with respect to y. The integral of the difference of the curves gives the area, which, after integration and evaluation, provides the result.

Step-by-step explanation:

To find the area between the curves x = y - y² and x = -3y² by integrating with respect to y, we need to set up the integral for the area by subtracting the second equation from the first. This will give us the width of a thin horizontal strip between the two curves at a given y value. We need to find the boundaries where the two curves intersect, which will be the limits of integration for y.

First, we equate the two equations to find the intersection points: y - y² = -3y². This simplifies to y² - y = 0, yielding solutions y = 0 and y = 1. These will be our limits of integration.

Next, we form the integral ∫₀¹ (y - y² - (-3y²)) dy, which simplifies to ∫₀¹ (y + 2y²) dy. Performing the integration, we get [rac{1}{2}y² + rac{2}{3}y³]₀¹, and upon evaluating this from 0 to 1, we find the area region.

Answer 2

0.0417 unit^2.

Step-by-step explanation:

First find the points at which the curves intersect

x = y - y^2

x = -3y^2

---> y - y^2 = -3y^2

---> 2y^2 + y = 0

---> y(2y + 1)= 0

y = -0.5, 0.

At these values x = -0.75 and 0.

The points of intersection are (0, 0) and (-0.75, -0.5)

The required area

-0.5

= ∫ -3y^2 - ∫y - y^2

0

= [ -y^3 - (y^2/2 - y^3/3)] between limits -0.5 and 0

= [0.125 - ( 0.125 - (-0.125/3)]

= -0.0417

We take the positive value 0.0417.