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.
Related Maths A Level answers
All answers ▸Solve int(ln(x)dx)
Why does exp(x+y) NOT equal exp(x)+exp(y)? [A-level Maths and Further Maths common mistake]
What is the product rule and when do you use it?
For the function f(x) = 4x^3 -3x^2 - 6x, find a) All points where df/dx = 0 and b) State if these points are maximum or minimum points.