How would I find the approximate area enclosed by the expression e^x*sin(x)*x^3 on an infinite scale?
The area under a curve is analytically calculated using the integral of the function. The integral of the function above could be calculated using integration by parts twice, considering that 3 functions are multiplied together, this could messy and a bit tricky. To work out an approximate area the shapes of the individual graphs of e^x, sin(x) and x^3 can be considered individually.
Sin(x) oscillates between 1 and -1 continuously, meaning that the area under the curve above and below the x axis will be approximately equal and opposite (positive for above the x axis and negative for below) on an infinite x axis resulting in the area under the curve being approximately zero.
The same goes for the graph of x^3. Where x is positive so are the y coordinates, where x is negative the y coordinates follow suit, meaning that the areas above and below the x axis will be approximately equal again, cancelling one another out.
Therefore, the only integral that actually needs to be considered is the area under y=e^x, which is y=e^x.
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