Question

Consider the area between the graphs x+4y=14 and x+7=y^2. This area can be
computed in two different ways using integrals.

First of all it can be computed as a sum of two integrals

They ask to use two integrals so i put f(x) from -7 to 2 which is correct
but for g(x) i put 2 to 14 for some reason 14 is wrong. I also put
f(x)=sqrt(x+7) and g(x)= (14-x)/4 and both are wrong wrong. I got
everything else correct except for these and I don't what I did wrong.

Answer 1

first, graph the two equations (in one cartesian plane)
then get the points of intersection:
x+7=y^2 *this is the second equation
x=y^2-7
substitute this to the first:
y^2-7+4y=14
y^2+4y-21=0
(y+7)(y-3)=0
y=-7 and y=3
substitute these back to obtain corresponding values of x:
*if y=-7,
x=(-7)^2-7=42
*if y=3,
x=(3)^2-7=2
therefore, points of int are (42,-7) and (2,3)

looking at the graph, i suggest you do vertical strips (that is, dx),, divide the whole area into region 1 (left side) and region 2 (right side),, after you do this, get the boundaries.
Region 1:
for x: the boundaries are the graph of the parabola (that is x=y^2-7) and the x-coord of the first point of int (x=2)
for y: since parabola is symmetric with respect to x-axis, y is from -3 to 3.

the area of region 1 is:
integral[from -3 to 3](integral[from y^2-7 to 2] dx)dy)
*note: this is double integral since A=dxdy

Region 2:
for region 2, i suggest you do horizontal strips (that is, dy) ,,then get the boundaries:
for x: from x-coord of first point of int (x=2) to x-ccord of 2nd point of int (x=42)
for y: from the parabola (y=sqrt(x+7)) to the line (y=(14-x)/4)

the area of region 2 is:
integral[from 2 to 42](integral[from sqrt(x+7) to (14-x)/4] dy)dx)

i'll leave the integration calculation to you,, add the areas and you'll finally get the whole area.

so there,, sorry for long explanation,,
i hope i was able to help,, =)