Solving an Integral Equation on a Curve C

In summary, the integral of the given curve, C, for the equation xy^2dx+(x+y)dy is 18538/5. However, it is important to be careful with notation and include all necessary parentheses.
  • #1
iRaid
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Homework Statement


Let the curve C be given by ##\vec{r}(t)=3t^{2}\hat{\imath}-\sqrt{t}\hat{\jmath}## between ##0 \leq t \leq 4##. Calculate ##\int_{C} xy^{2}dx+(x+y)dy##.


Homework Equations





The Attempt at a Solution


First find the derivative of r:
$$\vec{r}'(t)=6t\hat{\imath}-\frac{1}{2\sqrt{t}}\hat{\jmath}$$
$$\int_{C} xy^{2}dx+(x+y)dy=\int_{0}^{4} xy^{2}\hat{\imath}+(x+y)\hat{\jmath} \cdot (x'\hat{\imath}+y'\hat{\jmath})dt$$
$$\int_{0}^{4} (3t^{2})(\sqrt{t})^{2}\hat{\imath}+(3t^{2}-\sqrt{t})\hat{\jmath} \cdot (6t\hat{\imath}-\frac{1}{2\sqrt{t}}\hat{\jmath})dt$$
$$\int_{0}^{4}(3t^{3})(6t)+(3t^{2}-\sqrt{t})(\frac{-1}{2\sqrt{t}})dt=\int_{0}^{4} 18t^{4}+\frac{3t^{2}}{2\sqrt{t}}+\frac{1}{2}dt$$
$$=\frac{18t^{5}}{5}+\frac{\frac{3}{2}t^{\frac{5}{2}}}{\frac{5}{2}}+ \frac{t}{2}$$
Evaluate from 0 to 4.
=18538/5

I feel like I did something wrong...
 
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  • #2
Your answer is correct.

I am assuming it is just a typo, but be careful with your notation. Your missing some parentheses.

$$\int_{0}^{4} ((3t^{2})(\sqrt{t})^{2}\hat{\imath}+(3t^{2}-\sqrt{t})\hat{\jmath}) \cdot (6t\hat{\imath}-\frac{1}{2\sqrt{t}}\hat{\jmath})dt$$
 
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1. How do you solve an integral equation on a curve C?

To solve an integral equation on a curve C, you first need to express the equation in terms of a single variable. Then, you can use techniques such as substitution, integration by parts, or trigonometric identities to simplify the equation. Finally, you can use the fundamental theorem of calculus to evaluate the integral and find the solution.

2. Can any curve C be used to solve an integral equation?

Yes, any continuous curve C can be used to solve an integral equation. However, the complexity of the curve may affect the difficulty of the integral and the techniques needed to solve it.

3. What are some common techniques used to solve integral equations on a curve C?

Some common techniques used to solve integral equations on a curve C include substitution, integration by parts, trigonometric identities, and the fundamental theorem of calculus. Different techniques may be more suitable for different types of equations and curves.

4. Are there any limitations to solving integral equations on a curve C?

There are some limitations to solving integral equations on a curve C. For example, the curve must be continuous and the equation must be expressible in terms of a single variable. Additionally, some complex curves may require advanced mathematical techniques to solve the integral.

5. Can integral equations on a curve C be solved numerically?

Yes, integral equations on a curve C can be solved numerically using methods such as numerical integration or Monte Carlo simulations. These methods may be used when an analytical solution is not possible or when the integral is too complex to solve by hand.

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