Integral Evaluation: Evaluate the Sum of a Square Root

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In summary, the conversation discusses how to evaluate an integral by interpreting it in terms of areas. The integral is split into two parts, one representing a rectangle and the other representing a part of a circle. By graphing the integrand, the areas can be easily calculated without using calculus or Riemann sums.
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Homework Statement



Evaluate the integral by interpreting it in terms of areas.
[tex]\int(1+\sqrt{9-x^{2}})dx}[/tex]
The integral is from -3 to 0. I should be able to evaluate it as a limit of sums, since I've not been taught the fundamental theorem of calculus yet.

Homework Equations



dx=[tex]\frac{3}{n}[/tex], [tex]x_{i} = -3 + \frac{3i}{n}[/tex]

The Attempt at a Solution


I've gotten as far as [tex]3 + \frac{3}{n}\sum\sqrt{9-x^{2}[/tex] by applying summation rules. I just don't know how to evaluate a sum of a square root.

Thanks!
 
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  • #2
Evaluate the integral by interpreting it in terms of areas.

You're making this harder than it needs to be. Split the integral into two parts:
[tex]\int_{-3}^0 1 dx + \int_{-3}^0 \sqrt{9 - x^2}dx[/tex]

The region in the first integral is just a rectangle, so you should be able to get its area very easily. The second region is part of a circle. Can you figure out where the center of this circle is, its radius, and how much of the circle is represented by the integral? If so, you can evaluate this integral without using any calculus and without using Riemann sums.
 
  • #3
Since the question asks for evaluation by areas, graph the integrand from -3 to 0 and look at the resulting shape. Don't worry about summation.
 
  • #4
Ahh I get it now, thank you.
 

Related to Integral Evaluation: Evaluate the Sum of a Square Root

1. What is integral evaluation?

Integral evaluation is a mathematical process used to find the value of an integral, which is the area under a curve on a graph. It involves finding the antiderivative of a function and then evaluating it at the given limits of integration.

2. How do you evaluate the sum of a square root?

To evaluate the sum of a square root, you first need to simplify the expression by factoring out any perfect squares. Then, you can use the power rule for integrals to evaluate the expression. This involves adding 1 to the exponent and dividing by the new exponent. Finally, you can plug in the limits of integration and solve for the value of the integral.

3. What is the power rule for integrals?

The power rule for integrals states that the integral of x^n is equal to x^(n+1)/(n+1), where n is any real number except for -1. This rule is used to evaluate integrals of polynomial functions.

4. Can you use substitution to evaluate integrals involving square roots?

Yes, substitution can be used to evaluate integrals involving square roots. This involves substituting a new variable for the expression inside the square root and then solving for the integral using the power rule. This method can be useful for more complex integrals involving square roots.

5. What are some real-world applications of integral evaluation?

Integral evaluation has many practical applications in fields such as physics, engineering, and economics. It is used to calculate areas, volumes, and other quantities in real-world situations, such as finding the velocity of a moving object or the amount of product produced in a manufacturing process.

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