Integral Evaluation: Evaluate the Sum of a Square Root

[PsychStudent]

Homework Statement

Evaluate the integral by interpreting it in terms of areas.
$$\int(1+\sqrt{9-x^{2}})dx}$$
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=$$\frac{3}{n}$$, $$x_{i} = -3 + \frac{3i}{n}$$

The Attempt at a Solution

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

Thanks!

 

[Mark44]

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:
$$\int_{-3}^0 1 dx + \int_{-3}^0 \sqrt{9 - x^2}dx$$

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.

 

[mutton]

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.

 

[PsychStudent]

Ahh I get it now, thank you.