Solve for the Area of a Region: Integral of sqrt(9-x^2) over [0,3]

[Emethyst]

Homework Statement

Find the integral of sqrt(9-x^2) over [0,3]. You will not be able to find an antiderivative, so instead interpret the definite integral as the area of a region and compute the area geometrically (I haven't reached integration by substitution and integration by parts in class yet).

Homework Equations

The part I'm lost on

The Attempt at a Solution

This question has me stumped. I tried using both riemann sums and the trapezoid method but this didn't get me anywhere, as the answer is supposed to be 9pi/4. It is only out of 1 mark, so I know it can't be that difficult, but I'm still lost over it. Any pointers in the right direction here would be greatly appreciated. Thanks in advance.

 

[Pyrrhus]

Are you familiar with this geometry figure y^2 + x^2 = 3^2 ? Now consider what the square root does to this relation? (this is not a function), but when y = \sqrt{3^2 - x^2} what happens? (think in terms of Real value \sqrt{x} function)

 

[larstuff]

Homework Statement

Find the integral of sqrt(9-x^2) over [0,3]. You will not be able to find an antiderivative, so instead interpret the definite integral as the area of a region and compute the area geometrically (I haven't reached integration by substitution and integration by parts in class yet).

Homework Equations

The part I'm lost on

The Attempt at a Solution

This question has me stumped. I tried using both riemann sums and the trapezoid method but this didn't get me anywhere, as the answer is supposed to be 9pi/4. It is only out of 1 mark, so I know it can't be that difficult, but I'm still lost over it. Any pointers in the right direction here would be greatly appreciated. Thanks in advance.

Try downloading the program Geogebra (Web Start) - it's free math software, then let it draw the graph of this "weird" thing. You'll probably see what the answer is..

 

[Emethyst]

No I have not heard of that geometric figure before, but I do know that the square root prevents the function from crossing zero and becoming a negative number, and in a sense resembles half of a horizontal parabola. Now for the obvious question, how does that help me?

 

[Redbelly98]

Okay, how about

x² + y² = r²​

Is that figure more familiar to you?

 

[Emethyst]

Ohh it's a circle, I see it now, the radius is 3 so I just need to use the area formula and divide the answer by 4. Thanks for all the help guys