Answer to Question #185147 in Calculus for Angelo

Question

Find the area of the region R bounded by the given curves.

1. R: y = 4x and y = x^3
2. R: y = 6x - x^2 and y = x^2 - 2x
3. R: x^2 + 3y = 4 and x - 2y = 4
4. R: x + 2y = 2,y - x = 1 and 2x + y = 7

Answer 1

Area of regions bounded by curves

1.     y₁ = 4x and y₂ = x³ 

The two curves are equal at the point of intersection 

To get coordinates where the line and the curve intersect, we substitute  values of x and y to any of the two equations as follows

We have two regions, one at the first quadrant and the second at the third quadrant

The two curves are equal at the point of intersection 

To get coordinates where the line and the curve intersect, we substitute  values to any of the two equations as follows

3. x² + 3y= 4 and x-2y=4

Rearranging the equations and making the subject of the formula we have 

and making y the subject of the formula, we have

The two curves are equal at the point of intersection 

To get coordinates where the line and the curve intersect, we substitute  x values to any of the two equations as follows

Rearranging the equations and making the subject of the formula we have 

The three curves are equal at the point of intersection 

To get coordinates where the line and the curve intersect, we substitute x values to the three equations as follows