Use green's theorem to compute the area inside the ellipse x252+y2172=1. use the fact that the area can be written as ∬ddxdy=12∫∂d−y dx+x dy . hint: x(t)=5cos(t). the area is 85pi .

b. find a parametrization of the curve x2/3+y2/3=42/3 and use it to compute the area of the interior. hint: x(t)=4cos3(t).

The area of the ellipse is given by

To use Green's theorem, which says

( denotes the boundary of ), we want to find and such that

and then we would simply compute the line integral. As the hint suggests, we can pick

The line integral is then

We parameterize the boundary by

with . Then the integral is

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Notice that kind of resembles the equation for a circle with radius 4, . We can change coordinates to what you might call "pseudo-polar":

which gives

as needed. Then with , we compute the area via Green's theorem using the same setup as before:

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be

i do not know the answer

step-by-step explanation: