Plane Area in Rectangular Coordinates

Question

Find the area bounded between curves:

1.) x^2 = y + 4 and the x – axis from x = - 1 and x = 3.

2.) y = ( x + 1 )^3 and y = 0 from x = - 2 to x =1.

3.) 4x – 3y + 19 =0, y-axis, the lines y = 1 and y = 5.

4.) x = y^2 – 2y and x = 4y – y^2

Answer 1

1.) x^2 = y + 4 and the x – axis from x = - 1 and x = 3.

rewrite the equation as

y = x² - 4

this is a parabola shifted downward

*you have to graph it to understand

to find the x intercepts let y=0

0 = x² - 4

x² = 4 ,

so at x = 2 and at x= -2 the curve intersect the x axis.

To find the area from -1 to 3, you have to integrate from -1 to 2 then from 2 to 3

for the first integral, y will be the difference between y value of the x-axis

(which is 0 )to the y of the curve which is (x² - 4 ),

for the second integral y will be from the y of the curve to the x- axis

^{2 3}

∫ _-1 - ( x² - 4 ) dx + ∫₂ ( x² - 4) dx

and it is easy for you to complete the integration