# Area Under Curve: Integrating sqrt(36-.22x^2)

• ngigs
In summary, the area under the curve is a mathematical concept used to calculate the total area between a function and the x-axis within a specified interval. It is calculated by taking the integral of the function over the interval, with the variable x representing the independent variable in the function. The area under the curve cannot be negative and the square root in the function indicates that it is a curve rather than a straight line.
ngigs
how do you find the integral of the equation sqrt(36-.22x^2) between x=0 and x=9

Well, we have:
$$\int_{0}^{9}\sqrt{36-0.22x^{2}}dx=6\int_{0}^{9}\sqrt{1-(\frac{\sqrt{0.22}x}{6})^{2}}dx$$

Set:
$$u=\frac{\sqrt{0.22}x}{6}}\to{dx}=\frac{6}{\sqrt{0.22}}$$
And our integral may be rewritten as:
$$\frac{36}{\sqrt{0.22}}\int_{0}^{\frac{3\sqrt{0.22}}{2}}\sqrt{1-u^{2}}du$$

Do you have any ideas how to proceed from here?

## 1. What is the purpose of calculating the area under the curve?

The area under the curve is a mathematical concept that is used to calculate the total area between a function and the x-axis within a specified interval. It can be used to determine the total amount of a quantity, such as distance or volume, represented by the function.

## 2. How is the area under the curve calculated?

The area under the curve is calculated by taking the integral of the function over the specified interval. In this case, the integral of sqrt(36-.22x^2) would be taken from the lower limit to the upper limit of the interval.

## 3. What does the variable x represent in the function sqrt(36-.22x^2)?

The variable x represents the independent variable in the function. It is the value that is input into the function to determine the corresponding output or y-value.

## 4. Can the area under the curve be negative?

No, the area under the curve cannot be negative. The area under the curve is always a positive value, as it represents the total area between the function and the x-axis.

## 5. What is the significance of the square root in the function sqrt(36-.22x^2)?

The square root in the function indicates that the function is a curve, rather than a straight line. It also affects the shape of the curve and how it changes with different values of x.

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