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In summary, the conversation is about a linear integral involving exponential and trigonometric functions, and the use of Green's theorem to simplify the problem. The final integral involves a substitution and the use of standard techniques to solve it.
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Alexstrasza
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Homework Statement
I have a linear integral (e^xsiny-2)dx + (e^xcosy+x^2)dy
y≥0
2x=x^2+y^2
I used Green's theorem and got:
∬ (e^xcosy+2x) - (e^xcosy) dy dx
x bounds: from 0 to 2
y bounds: from 0 to sqrt(2x-x^2)
After solving all that stuff I get to:
∫ (2x) (sqrt(2x-x^2)) dx
x bounds: 0 to 2
I don't know how to calculate ∫ (2x) (sqrt(2x-x^2)) dx, I thought maybe I can turn the 2x into a derivative of 2x-x^2 but then I need to add +2 to the 2x and not sure if it is allowed?
I used an online integral calculator and it solved (answer: pi?) but the way of the solution was very very long so I think there must be an easier way to solve this (it is a test question).
Help please! ❀
Homework Equations
Calc II
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Alexstrasza said:
Homework Statement
I have a linear integral (e^xsiny-2)dx + (e^xcosy+x^2)dy
y≥0
2x=x^2+y^2I used Green's theorem and got:
∬ (e^xcosy+2x) - (e^xcosy) dy dx
x bounds: from 0 to 2
y bounds: from 0 to sqrt(2x-x^2)After solving all that stuff I get to:
∫ (2x) (sqrt(2x-x^2)) dx
x bounds: 0 to 2I don't know how to calculate ∫ (2x) (sqrt(2x-x^2)) dx, I thought maybe I can turn the 2x into a derivative of 2x-x^2 but then I need to add +2 to the 2x and not sure if it is allowed?
I used an online integral calculator and it solved (answer: pi?) but the way of the solution was very very long so I think there must be an easier way to solve this (it is a test question).
Help please! ❀
Homework Equations
Calc II
Write ##2x = (2x-2) + 2##, to get
$$\int_0^2 2x \sqrt{2x-x^2} \: dx = \int_0^2 (2x-2) \sqrt{2x- x^2} \: dx + 2 \int_0^2 \sqrt{2x - x^2} \: dx.$$
The first integral is easy; the second one is also standard, and you can look it up or consult a textbook for the appropriate changes of variables.
Last edited:
1. What is an integral?
An integral is a mathematical concept used in calculus to find the area under a curve. It is a fundamental tool for solving problems involving rates of change and accumulation.
2. Why do I get stuck on integrals?
Integrals can be challenging because they require a combination of algebraic and geometric manipulation. They also involve multiple steps and require a deep understanding of calculus concepts.
3. How can I improve my skills in solving integrals?
One way to improve your skills in solving integrals is to practice regularly. You can also consult with a tutor or study with a group to get a better understanding of the concepts and techniques involved.
4. What are some common techniques for solving integrals?
Some common techniques for solving integrals include substitution, integration by parts, and partial fractions. It is also helpful to have a good understanding of trigonometric and logarithmic functions.
5. What are some real-life applications of integrals?
Integrals have many real-life applications, such as calculating the volume of a shape, finding the center of mass of an object, and determining the work done by a force over a distance. They are also used in physics, engineering, and economics.
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