Region enclosed by circle and line help
Need some help on a calculus assignment. Completely stumped and my long distance lecturers are non existent (as always) and my study guide doesn't have enough examples to be useful for this question.
So the full question is
Let R be the Region enclosed in the circle $x^2+y^2=1$ and below the line $y=x+1$
(a) Sketch the Region R and find the points of intersection (see image below)
b) Is R a Type I region? If yes describe it as a Type I region. If not , describe it as a union of Type I regions. Use set builder notation
c) Same as question b but uses Type II region
Now I've not been able to answer b) or c). This is either because my region is wrong or because I don't understand how the region can be Type I or Type II without it being a union in set builder notation
PS: Should I be thinking of using polar coordinates? The reason I think not is the question is identified as being part of a section of work that excludes the polar coordinates, so I think it should be ignored for now (unless it's the only solution)
$\endgroup$ 61 Answer
$\begingroup$Type I: The question is can the region be described as a region between two constants, and two functions over $x$, or a union of such regions. Is the region of Type I? No, so let's find a union:
Separate the region along the line $x=0$ and call the regions $D_1$ and $D_2$.
$D_1$ can be described as the region with the boundaries of $f\left(x \right) = x+1$ and $g\left(x \right) = -\sqrt{ \left(1-x^2 \right)}$ on the domain of $-1\le x \le 0$
$D_2$ can be described as the region with the boundaries of $h\left(x \right) = \sqrt{ \left(1-x^2 \right)}$ and $k\left(x \right) = -\sqrt{ \left(1-x^2 \right)}$ on the domain of $0\le x \le -1$
Both of these regions are of Type I.
The same logic follows for Type II.
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