Use Green's Theorem to evaluate the line integral. Orient the curve counerclockwise. oint_C F8dr, where F(x,y)=<<x^2,x^2>> and C consists of the arcs y = x^2 and y = 8x for 0 <= x <= 8

Mylo O'Moore 2020-10-20 Answered
Use Green's Theorem to evaluate the line integral. Orient the curve counerclockwise.
\(\displaystyle\oint_{{C}}{F}{8}{d}{r}\), where \(\displaystyle{F}{\left({x},{y}\right)}={\left\langle{x}^{{2}},{x}^{{2}}\right\rangle}\) and C consists of the arcs \(\displaystyle{y}={x}^{{2}}{\quad\text{and}\quad}{y}={8}{x}{f}{\quad\text{or}\quad}{0}\le{x}\le{8}\)

Want to know more about Green's, Stokes', and the divergence theorem?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Adnaan Franks
Answered 2020-10-21 Author has 16436 answers
Step 1
We have given the line integral with values,
\(\displaystyle{f{{\left({x},{y}\right)}}}={\left\langle{x}^{{2}},{x}^{{2}}\right\rangle}{\quad\text{and}\quad}{a}{r}{c}{s}{y}={x}^{{2}},{y}={8}{x}\)
Step 2
We know the green's theorem formula to calculate line integral,
\(\displaystyle\int_{{C}}{P}{\left.{d}{x}\right.}+{Q}{\left.{d}{y}\right.}=\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)
Now we shall plug all the values in the formula,
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\int_{{0}}^{{8}}}{\int_{{{x}^{{2}}}}^{{{8}{x}}}}{\left({2}{x}-{0}\right)}{\left.{d}{y}\right.}{\left.{d}{x}\right.}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\int_{{0}}^{{8}}}{\int_{{{x}^{{2}}}}^{{{8}{x}}}}{2}{x}{\left.{d}{y}\right.}{\left.{d}{x}\right.}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\int_{{0}}^{{8}}}{2}{x}{{\left[{y}\right]}_{{{x}^{{2}}}}^{{{8}{x}}}}{\left.{d}{x}\right.}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\int_{{0}}^{{8}}}{2}{x}{\left[{8}{x}-{x}^{{2}}\right]}{\left.{d}{x}\right.}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\int_{{0}}^{{8}}}{{\left[{16}{x}^{{2}}-{2}{x}^{{3}}\right]}_{{0}}^{{8}}}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={{\left[{16}\frac{{x}^{{3}}}{{3}}-\frac{{{2}{x}^{{4}}}}{{4}}\right]}_{{0}}^{{8}}}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\left[{16}\frac{{8}^{{3}}}{{3}}-\frac{{8}^{{4}}}{{2}}\right]}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\left[{16}\frac{{512}}{{3}}-\frac{{4096}}{{2}}\right]}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\left[{2730.66}-{2048}\right]}\)
\(\displaystyle\int\int_{{D}}\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={682.66}\)
Step 3
So the value of line integral is 682.66
Not exactly what you’re looking for?
Ask My Question
43
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2022-01-16
Brad and Lena are recording their classmates' eye color for a statistics assignment. In Brad's class, 3 out of the 25 students have green eyes. In Lena's class, 2 out of the 20 students have green eyes. Which class has the greater ratio of green-eyed students to total students?
asked 2022-01-17
As Mark Ellon is getting ready for a meeting, the room goes dark. He fishes for a green shirt in his drawer. He wears only blue, green and white colors. His drawers have identical shirts in these colors: 28 blue, 25 green, and 13 white shirts. How many shirts did he throw on the floor to be a cent percent sure that he has a taken out a green shirt?
asked 2022-01-18
The ratio of the number of blue sticks to the number of green sticks in a box was 4:1. When David took out some blue and sticks and replaced them with an equal number of green sticks, the ratio of the number of blue sticks to the number of green sticks became 3:1. If there were 185 green sticks in the box now, (a) find the total number of blue and green sticks in the box, (b) find the number of green sticks in the box at first.
asked 2022-01-18
There are 16 shirts in your closet, 6 blue and 10 green. You randomly select one to wear on Monday, do not return it to the closet, then select one to wear on Tuesday. What is the probability of wearing a blue shirt on Monday and a green shirt on Tuesday?
asked 2020-12-02
Evaluate the line integral \(\displaystyle\oint_{{C}}{x}{y}{\left.{d}{x}\right.}+{x}^{{2}}{\left.{d}{y}\right.}\), where C is the path going counterclockwise around the boundary of the rectangle with corners (0,0),(2,0),(2,3), and (0,3). You can evaluate directly or use Green's theorem.
Write the integral(s), but do not evaluate.
asked 2022-01-18
Find the conditional probability of the given event when two fair dice (one red and one green) are rolled.
The red one is 6, given that the green one is 6.
asked 2022-01-19
Randomly drawing three green skittles in a row from a bag that contains eight green skittles out of 25 skittles total.
...