91Ó°ÊÓ

Evaluate integral \(\oint_{C}\left(x^{2}+y^{2}\right) d x+2 x y d y,\) where \(C\) is the curve that follows parabola \(y=x^{2}\) from \((0,0)(2,4),\) then the line from \((2,4)\) to \((2,\) 0), and finally the line from \((2,0)\) to \((0,0).\)

Evaluate line integral \(\oint_{C}(y-\sin (y) \cos (y)) d x+2 x \sin ^{2}(y) d y, \) where \(C\) is oriented in a counterclockwise path around the region bounded by \(x=-1, x=2, y=4-x^{2},\) and \(y=x-2.\)

Use Green’s theorem to find the area. Find the area between ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) and circle \(x^{2}+y^{2}=25.\)

Find the area of the region enclosed by parametric equation \(p(\theta)=\left(\cos (\theta)-\cos ^{2}(\theta)\right) \mathbf{i}+(\sin (\theta)-\cos (\theta) \sin (\theta)] j\) for \(0 \leq \theta \leq 2 \pi.\)

Find the area of the region bounded by hypocycloid \(\mathbf{r}(t)=\cos ^{3}(t) \mathbf{i}+\sin ^{3}(t) \mathbf{j} .\) The curve is parameterized by \(t \in[0,2 \pi].\)

Find the area of a pentagon with vertices \((0,4),(4,1),(3,0),(-1,-1),\) and \((-2,2).\)

Use Green’s theorem to evaluate \(\int_{C+}\left(y^{2}+x^{3}\right) d x+x^{4} d y,\) where \(C^{+}\) is the perimeter of square \([0,1] \times[0,1]\) oriented counterclockwise.

Use Green's theorem to find the area of one loop of a four-leaf rose \(r=3 \sin 2 \theta .\left(\text { Hint: } x d y-y d x=\mathbf{r}^{2} d \theta\right).\)

Use Green’s theorem to find the area under one arch of the cycloid given by parametric plane \(x=t-\sin t, y=1-\cos t, t \geq 0.\)

Evaluate Green’s theorem using a computer algebra system to evaluate the integral \(\int_{C} x e^{y} d x+e^{x} d y,\) where \(C\) is the circle given by \(x^{2}+y^{2}=4\) and is oriented in the counterclockwise direction.