# The total area between 3sin(5x) and the x-axis

• B
• Saracen Rue
In summary, to calculate the total area between f(x) and the x-axis between the origin and any given intercept, we can use integration to find the area under the curve. However, we need to account for parts of the curve that are below the x-axis by using integration of the absolute value of the function. Additionally, we can use symmetry to simplify the calculation by only considering the area between (0,0) and (pi/5, 0), and then multiplying it by the number of peaks (n) over the total domain. This results in a final equation of total area = (5a/pi) * integration(5pi/3, 0, 3|sin(5x)|).

#### Saracen Rue

If you were given f(x) = 3sin(5x), would it be possible to express the total area between f(x) and the x-axis between the origin and any given intercept? Basically, could you form a general equation for the total area for f(x) where x∈[0,a] and a is an x-intercept.

Saracen Rue said:
If you were given f(x) = 3sin(5x), would it be possible to express the total area between f(x) and the x-axis between the origin and any given intercept? Basically, could you form a general equation for the total area for f(x) where x∈[0,a] and a is an x-intercept.
Integration gives area under the curve. You'll get the area if you calculate ∫3sin(5x)dx with limits 0 to a.

Saracen Rue said:
If you were given f(x) = 3sin(5x), would it be possible to express the total area between f(x) and the x-axis between the origin and any given intercept? Basically, could you form a general equation for the total area for f(x) where x∈[0,a] and a is an x-intercept.
Yes, but to get the area, you need to account for parts of the curve that are below the x-axis. For example, on the interval ##[\pi/5, 2\pi/5]## the graph of this function is below the x-axis. On intervals such as this one, the integral gives a negative value. To to get the area between the curve and the x-axis, you would need an integral like this: ##\int_{\pi/5}^{2\pi/5}-3\sin(5x)dx##

cnh1995 said:
Integration gives area under the curve. You'll get the area if you calculate ∫3sin(5x)dx with limits 0 to a.
This won't necessarily give the area. For example, ##\int_0^{2\pi/5}3\sin(5x)dx## evaluates to 0, but the area between the curve and the x-axis is 12/5.

You need to compute $\int 3|sin(5x)|dx$, which means dividing the domain of integration into pieces where x is a multiple of $\frac{\pi}{5}$.

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I have done some thinking about this an I think I've figured out a way to do it.

The graph intersects the x-axis every n*pi/5 interval. (i.e. 0*pi/5 = 0, 1*pi/5 = pi/5, 2*pi/5, 3*pi/5, etc)
Due to symmetry in the graph, we also know the area between the positive and negative sections of the graph will also be the same (i.e. the area between (0,0) to (pi/5, 0) and (pi/5, 0) to (2pi/5, 0) is the same, however integrating for the latter area results in a negative. However, this means if we just calculate the area over just (0,0) to (pi/5, 0) and multiply it by the number of areas the graph has over the given domain, we should get the total area.

The domain is [0, a], where a is an x-intercept. This means a = n*pi/5. As n also equals the number of areas up to that point (i.e. at 7pi/5, there have been 7 areas above/below the x-axis)
a = n*pi/5
n = 5a/pi
As discussed earlier, the total area should equal the area over the domain [0, pi/5] multiplied by the number of peaks (n) over the total domain. Total area = n * integration(5pi/3, 0, 3sin(5x)) = 5a/pi * integration(5pi/3, 0, 3sin(5x))

## What is the equation for the total area between 3sin(5x) and the x-axis?

The equation for the total area between 3sin(5x) and the x-axis is given by A = ∫(3sin(5x) - 0)dx, where A represents the total area and dx represents the infinitesimal change in the x-axis.

## What is the significance of the total area between 3sin(5x) and the x-axis?

The total area between 3sin(5x) and the x-axis represents the net displacement of the function 3sin(5x) from the x-axis. It can also be interpreted as the total positive and negative areas enclosed by the curve.

## How is the total area between 3sin(5x) and the x-axis calculated?

The total area between 3sin(5x) and the x-axis is calculated using the definite integral, which involves finding the antiderivative of the function and plugging in the upper and lower limits of integration. This gives the net area enclosed by the curve.

## Does the total area between 3sin(5x) and the x-axis change with different values of x?

Yes, the total area between 3sin(5x) and the x-axis changes with different values of x. This is because the function 3sin(5x) varies with different values of x, resulting in different net displacements from the x-axis and thus, different total areas.

## How can the total area between 3sin(5x) and the x-axis be visualized?

The total area between 3sin(5x) and the x-axis can be visualized using a graphing calculator or graphing software. The graph will show the curve of 3sin(5x) and the area enclosed between the curve and the x-axis. It can also be visualized by breaking the interval of integration into smaller intervals and calculating the area under each interval using rectangles or trapezoids.