Question

Consider the area between the graphs x+6y=9 and x+7=y^2...?

This area can be computed as a sum of two integrals: int. f(x)dx between a and b, and int. g(x)dx between b and c. What are the values of f(x), g(x), a, b and c?

Answer 1

Step 1:

The line equation is and the curve equation is .

First find the points of intersection.

and

and .

Substitute the values and solve for corresponding values of .

Points of intersections are and .

The parabola equation is .

Vertex of the parabola at .

Step 2:

Graph the region:

.

 

Answer 2

Contd...

Step 3:

The parabola equation is

The limits ,

Each vertical cross section is between top part of parabola and bottom part of parabola :

Height of cross-section

The limits ,

Each vertical cross section is between the line  of parabola and bottom part of parabola :

Height of cross-section .

In this case the  area is sum of two integrals is .

Substitute the values , ,

And , .

.

Solution:

.