Green's Theorem - AP Calculus BC
Card 0 of 15
Use Green's Theorem to evaluate
, where
is a triangle with vertices
,
,
with positive orientation.
Use Green's Theorem to evaluate , where
is a triangle with vertices
,
,
with positive orientation.
First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).


In this particular case
, and
, where
, and
refer to
.
We know from Green's Theorem that

So lets find the partial derivatives.









First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).
In this particular case , and
, where
, and
refer to
.
We know from Green's Theorem that
So lets find the partial derivatives.
Compare your answer with the correct one above
Use Green's Theorem to evaluate the line integral

over the region R, described by connecting the points
, orientated clockwise.
Use Green's Theorem to evaluate the line integral
over the region R, described by connecting the points , orientated clockwise.
Using Green's theorem
![\int_C{F}dr=\int \int\left[{\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766279/gif.latex)


since the region is oriented clockwise, we would have
![\int_C{F}dr=-\int\int\left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766282/gif.latex)
which gives us

Using Green's theorem
since the region is oriented clockwise, we would have
which gives us
Compare your answer with the correct one above
Use Greens Theorem to evaluate the line integral

over the region connecting the points
oriented clockwise
Use Greens Theorem to evaluate the line integral
over the region connecting the points oriented clockwise

Using Green's theorem
![\int_C F dr=\int\int \left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/768557/gif.latex)


Since the region is oriented clockwise


Using Green's theorem
Since the region is oriented clockwise
Compare your answer with the correct one above
Use Green's Theorem to evaluate
, where
is a triangle with vertices
,
,
with positive orientation.
Use Green's Theorem to evaluate , where
is a triangle with vertices
,
,
with positive orientation.
First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).


In this particular case
, and
, where
, and
refer to
.
We know from Green's Theorem that

So lets find the partial derivatives.









First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).
In this particular case , and
, where
, and
refer to
.
We know from Green's Theorem that
So lets find the partial derivatives.
Compare your answer with the correct one above
Use Green's Theorem to evaluate the line integral

over the region R, described by connecting the points
, orientated clockwise.
Use Green's Theorem to evaluate the line integral
over the region R, described by connecting the points , orientated clockwise.
Using Green's theorem
![\int_C{F}dr=\int \int\left[{\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766279/gif.latex)


since the region is oriented clockwise, we would have
![\int_C{F}dr=-\int\int\left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766282/gif.latex)
which gives us

Using Green's theorem
since the region is oriented clockwise, we would have
which gives us
Compare your answer with the correct one above
Use Greens Theorem to evaluate the line integral

over the region connecting the points
oriented clockwise
Use Greens Theorem to evaluate the line integral
over the region connecting the points oriented clockwise

Using Green's theorem
![\int_C F dr=\int\int \left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/768557/gif.latex)


Since the region is oriented clockwise


Using Green's theorem
Since the region is oriented clockwise
Compare your answer with the correct one above
Use Green's Theorem to evaluate
, where
is a triangle with vertices
,
,
with positive orientation.
Use Green's Theorem to evaluate , where
is a triangle with vertices
,
,
with positive orientation.
First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).


In this particular case
, and
, where
, and
refer to
.
We know from Green's Theorem that

So lets find the partial derivatives.









First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).
In this particular case , and
, where
, and
refer to
.
We know from Green's Theorem that
So lets find the partial derivatives.
Compare your answer with the correct one above
Use Green's Theorem to evaluate the line integral

over the region R, described by connecting the points
, orientated clockwise.
Use Green's Theorem to evaluate the line integral
over the region R, described by connecting the points , orientated clockwise.
Using Green's theorem
![\int_C{F}dr=\int \int\left[{\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766279/gif.latex)


since the region is oriented clockwise, we would have
![\int_C{F}dr=-\int\int\left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766282/gif.latex)
which gives us

Using Green's theorem
since the region is oriented clockwise, we would have
which gives us
Compare your answer with the correct one above
Use Greens Theorem to evaluate the line integral

over the region connecting the points
oriented clockwise
Use Greens Theorem to evaluate the line integral
over the region connecting the points oriented clockwise

Using Green's theorem
![\int_C F dr=\int\int \left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/768557/gif.latex)


Since the region is oriented clockwise


Using Green's theorem
Since the region is oriented clockwise
Compare your answer with the correct one above
Use Green's Theorem to evaluate
, where
is a triangle with vertices
,
,
with positive orientation.
Use Green's Theorem to evaluate , where
is a triangle with vertices
,
,
with positive orientation.
First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).


In this particular case
, and
, where
, and
refer to
.
We know from Green's Theorem that

So lets find the partial derivatives.









First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).
In this particular case , and
, where
, and
refer to
.
We know from Green's Theorem that
So lets find the partial derivatives.
Compare your answer with the correct one above
Use Green's Theorem to evaluate the line integral

over the region R, described by connecting the points
, orientated clockwise.
Use Green's Theorem to evaluate the line integral
over the region R, described by connecting the points , orientated clockwise.
Using Green's theorem
![\int_C{F}dr=\int \int\left[{\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766279/gif.latex)


since the region is oriented clockwise, we would have
![\int_C{F}dr=-\int\int\left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766282/gif.latex)
which gives us

Using Green's theorem
since the region is oriented clockwise, we would have
which gives us
Compare your answer with the correct one above
Use Greens Theorem to evaluate the line integral

over the region connecting the points
oriented clockwise
Use Greens Theorem to evaluate the line integral
over the region connecting the points oriented clockwise

Using Green's theorem
![\int_C F dr=\int\int \left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/768557/gif.latex)


Since the region is oriented clockwise


Using Green's theorem
Since the region is oriented clockwise
Compare your answer with the correct one above
Use Green's Theorem to evaluate
, where
is a triangle with vertices
,
,
with positive orientation.
Use Green's Theorem to evaluate , where
is a triangle with vertices
,
,
with positive orientation.
First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).


In this particular case
, and
, where
, and
refer to
.
We know from Green's Theorem that

So lets find the partial derivatives.









First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).
In this particular case , and
, where
, and
refer to
.
We know from Green's Theorem that
So lets find the partial derivatives.
Compare your answer with the correct one above
Use Green's Theorem to evaluate the line integral

over the region R, described by connecting the points
, orientated clockwise.
Use Green's Theorem to evaluate the line integral
over the region R, described by connecting the points , orientated clockwise.
Using Green's theorem
![\int_C{F}dr=\int \int\left[{\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766279/gif.latex)


since the region is oriented clockwise, we would have
![\int_C{F}dr=-\int\int\left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/766282/gif.latex)
which gives us

Using Green's theorem
since the region is oriented clockwise, we would have
which gives us
Compare your answer with the correct one above
Use Greens Theorem to evaluate the line integral

over the region connecting the points
oriented clockwise
Use Greens Theorem to evaluate the line integral
over the region connecting the points oriented clockwise

Using Green's theorem
![\int_C F dr=\int\int \left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/768557/gif.latex)


Since the region is oriented clockwise


Using Green's theorem
Since the region is oriented clockwise
Compare your answer with the correct one above