# Integral by interpreting it in terms of area

• tsukuba
In summary: Therefore, the area under this function from -3 to 0 should be the area of a quarter circle of radius 4 (since 1 is added to the front).In summary, the given integral has limits of integration from -3 to 0, and the function within the integral is 1 + sqrt(9 - x^2). By rearranging the equation, it can be seen that the function represents a quarter circle of radius 4, and the integral is finding the area under this curve.
tsukuba

## Homework Statement

∫(a= -3 , b= 0) (1 + √9 - x^2) dx

## Homework Equations

∫(a,b) f(x) dx = lim as n → $\infty$ $\sum$ f(xi) delta x

## The Attempt at a Solution

I tried plugging in my a and b value into the function just as I would with any other function to find the area and i get a number but the answer is a ∏ so I am not sure with the pi comes from

What did you plug your numbers into? That's not an elementary integral. I think the intention was for you to think about what the graph of 1 + sqrt(9 - x^2) looks like, and then use geometry to calculate it.

I've been doing some research and i figure that 9-x^2 is 1/4 of a circle

tsukuba said:
I've been doing some research and i figure that 9-x^2 is 1/4 of a circle
There is certainly a quarter circle involved, but the region is a bit more than that.

how do i know its a quarter circle though?

A couple different ways. One is to rearrange the equation a bit.

Replace f(x) with y, and ignore the + 1 in front, and you have this:
$$y = \sqrt{9 - x^{2}}$$
With a little rearranging (square both sides, move x^{2} over), you get this:
$$x^{2} + y^{2} = 9$$
You can recognize that as the equation of a circle, with radius 3. So if plotted at the origin, it would stretch from -3 to 3 on both axes. Since the limits on integration are from x = -3 to x = 0, that's half of the circle. Then, since you take the positive square root, it's the half of the circle above the x-axis: thus, a quarter circle in the second quadrant. Look at the addition of 1 to the front of the original equation (f(x)), and you get that same quarter circle moved up one.

1 person

## What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total value of a function over a given interval.

## How is an integral interpreted in terms of area?

An integral can be interpreted as the area under a curve on a graph. The x-axis represents the independent variable and the y-axis represents the dependent variable. The integral calculates the total area between the curve and the x-axis within a given interval.

## What is the significance of interpreting an integral in terms of area?

Interpreting an integral in terms of area allows us to visualize and understand the relationship between a function and its integral. It also helps us to calculate the total value of a function, which can have practical applications in various fields such as physics, engineering, and economics.

## What are the different methods of calculating an integral in terms of area?

The most common methods of calculating an integral in terms of area include the Riemann sum, the Trapezoidal rule, and Simpson's rule. These methods involve dividing the area under the curve into smaller, simpler shapes and using different formulas to calculate their areas.

## Can an integral be negative?

Yes, an integral can be negative. This can occur when the function being integrated has negative values within the given interval. In this case, the area under the curve will be subtracted from the total area, resulting in a negative value for the integral.

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